Surfing on the ocean of numbers - a Few Smarandache Notions and Similar Topics

Surfing on the ocean of numbers -a Few Smarandache Notions and Similar Topics Hervy Ibstedt

Emus University Press Vail

1997

Surfing on the ocean of numbers -a Few Smarandache Notions and Similar Topics Henry Ibstedt

•

Erhus University Press Vail

1997

Surfing on the ocean of numbers -a Few Smarandacbe Notions and Similar Topics Henry Ibstedt Glimminge 2036 280 60 Broby Sweden (May - October) 7, rue du Sergent Blandon 92130 Issy les Moufineaux France (November-April)

Erbus University Press Vail 1997

Š Henry Ibstedt & Erbus University Press

The cover picture refers to the last article in this book. It illustrates cumulative statistics on the occurrence of square free integers with 1, 2, 3, etc. prime factors for successive intervals of integers. A detailed discussion is given in the article

Printed in the United States of America

by Erhus University Press 13333 Colossal Cave Road Vail, Box 722, AZ 85641, USA E-mail: Research37.@aol.com Comments on our books and journals are welcome. Manuscript, for publication may be sent to the above address for consideration.

ISBN 1-879585-S7-X Sta..'ldard Address N\IlIlber 297-5092

To the Memory of my Son Carl-Magnus

Preface

Surfing on the Ocean of Numbers - why this title? Because this little book does not attempt to give theorems and rigorous proofs in the theory of numbers. Instead it will attempt to throw light on some properties of numbers, nota bene integers, through a study of the behaviour of large numbers of integers in order to draw some reasonably certain conclusions or support already made conjectures. But no matter how far we extend our search or increase our samples in these studies we are in fact, in spite of more and more powerful technologies, merely skimming the surface of the immense sea of numbers. - Hence the title.

Most books in Mathematics are used as reference books. I still consult my fIrst Number Theory book which I bought in 1949 - Elementary Number Theory by Uspensky and Heaslet. It was when I was yOilllg and enthusiastic and dreamt about becoming a Mathematician. I am still enthusiastic but I became a Physicist instead..

However, I stayed on the theoretical side and avoided to have to much to do with things that can break. But even so experiments are a major source of knowledge and maybe this book shows a little of a Physicist's approach to Mathematics. Most results are presented or supported by tables and graphs. All calculations have been carried out on a Pentium 100 Mhz laptop using Ubasic as a programming language. Finally, the author has tried to make a book which should be easy and pleasant to read.

A word about the beauty of Mathematics and Number Theory in particular. The crystallized truth of a theorem, where a whole spectrum of mathematical thoughts come together to form an entity, is like a painting where designs and colours merge into a work of art. But sometimes it is not the fInished result which is the most interesting - it could be the unsolved problem itself. Why? Maybe it is the challenge of getting somewhere with it or the hours and days of thinking and trying that occupy the mind in a positive sense different from the problems of our time. It all brings piece to the mind - it's like walking in the silence of the forest enjoying the trees, the sun and the blue sky , and should it happen that all the bits and pieces suddenly fall into place to give a solution then it is the most sublime experience for the human mind - eureka!

But then the interest in the problem fades away unless solving the problem created new ones - and that is almost always the case.

5

Most topics in this book have been selected from Only Problems, Not Solutions by F. Smarandache. Others have bee suggested by Dr. R. Muller of Erhus University Press.

A few problems which the author has fotmd interesting originate from the Numbers Count Colwnn of Personal Computer World. This journal has had great importance for the author as a source of recreational Mathematics and I take this opportunity to thank the Editor of this colwnn Mike Mudge for all correspondence and encouragement he gave me in the past. lllustrations, graphics, layout and [mal editing up to camera ready torm has been done by the author. Tables have been created by direct transfer from computer tiles established at the time of computation to the manuscript so as to avoid t)ping error s.

This book has come into being thanks to R. Muller at Erhus University Press who has never failed an opportunity to give his support and encouragement. Rapid e-mail exchange between him in the USA and me in France has greatly facilitated our work. I also thank Dr. Muller's colleagues for their help. Many thanks are also due to my son Michael Ibstedt for his help and advice concerning computer equipment and software.

Last but not least my warm thanks to my dear wife Anne-Marie for her encouragement and endless patience with a husband who does not always listen because his mind is somewhere else.

February 1997 Henry Ibstedt

Contents L

On Prime Numbers The sequence a路Pn + b Prime Number Gaps

9 13

II. On Smarandache Functions Smarandache-Fibonacci Triplets Radu's Problem The Smarandache Ceil Function The Smarandache Pseudo Function Z(n)

19

24 27 30

ill. Loops and Invariants Perfect Digital Invariants and Related Loops The Squambling Function Wondrous Numbers Iterating d(n) and (J(n) - Two Problems proposed by F. Smarandache

39 45 46

52

IV. Diophantine Equations Some Thoughts on the Equation I).p - xq I=k The Equation 7(p4+q4+r4+s4+t4)_5(p2+q2+r+s2+t2i+90pqrst=0 The Equation y=2路XIX2 ... Xk+ 1 7

59 68 70

Chapter I On Prime Numbers

This chapter deals with some computational observations on prime numbers and their distribution. These computations only skim the surface of the ocean of integers but they give an idea of the general behavior of primes and often support some of the many conjectures that are made concerning primes. Computer programs have been written in Ubasic with extensive use of some of the built in functions of this language.

In some cases the use of these functions will be illustrated with a few lines of program code. Most results are given in tabular and/or graphical term.

1.

On the Sequence a-pD + b

In his book Only Problems, Not Solutions} F. Smarandache asks the tellowing question: If (a,b)=l, how many primes does the progression a·Pn + b, where Pn is prime and nf:{l,2, ... }, contain" Already for a=l and b=2 we run into the classical unsolved problem "Are there infinitely many twin primes? ". The answer to how manv? is certainly equally difficult for other sets of parameters a,b. However, some interesting information on how a·Pn + b behaves will be obtained for the fIrst 10,000,000 primes Pn. Let m be the number of primes produced by a·Pn + b for ~, i.e. if a·Pn + b is prime we can write a·Pn + b = qm where qrn is prime. The following Ubasic program lines have been used to determine whether a·Pn + b is prime or not: while N<lOOOOOOI p=nxtprm(p) incN c=a%*p+b% if nxtprm( c-l )=c then inc m wend The program has been implemented for a set of values of the parameters a and b. The result is shown in table 1. It is interesting to visualize the result. Because of the logarithmic behaviour of the distribution of primes it is reasonable represent mIN as a

I

Cnsolved problem number 17.

9

function of 10glON rather than as a function of N. For this reason the value of m has been recorded during the computation for N=lO, 10 2, 103, 104, 10 5, 106, and 10 7

Table 1. Number of primes m in the progression a,blN 1,2 2,1 3,2 4' 5,2

.),1 7.2

10 5 5 8 3 5 7 4

1()3

102 25 25 47 21 26 39 23

10' 1270 1221 2350 1108 1492 2175 1288

17A I~

166 290 145 188 277 167

a,pr~b

lOS

10250 9667 1891" 9314 12020 ,8019 10634

for n<N

1()6 86027 82236 160127 78676 103010 153925 91232

lCY

738597 711153 1392733 685069 903165 1342255

The graphs in figure I show mIN (y-axis) as a function oflog]():'l" (x-axis) ""nere m is the number of primes of the form aPn+b for n<N. Figure lb is an enlargement of figure la for large values ofN, (N :5:10 7 ). The eight curves correspond to the following sets of

0&-----------

o o.

o

,

a

)~--------------------------~ 1

6

3

Figure lao

10

7

o.~-----------------------------------------------------,

0t-

--'"'-"'---....-.;:---_._---------_...._---

I

1 ...J

5.5

35

6

"1

6.S

Figre lb.

paE. .,-,5 listed in the order in. which the curves appear from top to bottom in the right '":d side of the two figcres: (a,b)=(3,2), (6,1), (5,2), (7,2), (1,2) (2,1), (4,1) and (8,1) Tabie 2. Number of crimes

biN

10

2

5 4

4

6

8 10

12

., I

5 5 7

"!

IT' ge~e~ated

路:C

iCXJC

25 27

174

48

344 178 231 340

~70

24 34 48

11

oy pn+b for rs;N

lCCXXJ 1270 1264 2538 1303 1682 2515

': O~v:::tJ

~~'Y:):)):

le250

86:>27

1Q21A

85834 170910 85866

20472 10336 13653 20462

114394

171618

b=2

b=8

O..'j

05

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0 4

2

6

2

3

O.S

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 4

3

6

2

0.4

r\.

5

6

b=12 0.7

,

0.5

"'

0.2 0.1

0.6

:

'\.

0.3

o

4

3

b=6

O.S

6

0 2

0.6

S

b=1O

b=4 O.S

0.7

4

0.4 ~.

-

r\.' '\.

"'-

"- ~

0.3 0.2

,

-

5

6

,

0.1 0

2

3

4

2

6

3

Rgure 2. The ratio m/M plotted against logN for b=2,4,6,8,1O,and 12

12

4

The nwnber of primes m in the sequence a路p,,+b for n = 1, 2, ... N is illustrated in figure 2 for a = 1 and b = 2, 4, 6, 8, 10 and 12 where the ratio M=m/N is plotted against 10gN. The corresponding nwnerical results are given in table 2.

The general appearance of the graphs for various values of the parameters (a,b) in ap,,+b is very similar. In particular figure lb shows an interesting picture of curves running parallel to one another and in particular to the one for (a,b)=(l,2), that is the curve for p+2 which corresponds to prime twins for which we have the classical conjecture that there are infinitely many. This makes the following conclusion reasonable.

Conjecture: The nwnbers.

2.

progression ap,,+b , (a,b)=l contains infinitely many prime

Prime Number Gaps

Smarandache asked how many primes there is in the progression ap,,+b. For a=l and b=2 the question is equivalent to 'how many twin primes are there?'. Since we have a very stable conjecture that there are infinitely many we now want to know something about their distribution and also about the distribution of other prime nwnber gaps g=p"..l-P". With a small change in the Ubasic program used in the previous section we can study the distribution of primes over gaps g=2, 4, 6, ... p=3 whilep<N q=p p--nxtprm(p)

u=(p-q)/2 inc ftu) ifp(u)=O then p(u)=q wend

'Count the nwnber of gaps = p-q.

'Store the smallest prime for which the 'gap occurs in pc).

This program was run for primes p<N=2路l 09 The result is shown in table 3,where f is the nwnber of gaps g and p the prime nwnber for which the gap first occurs, N<2路1Q7.

All gaps ~92 except 264, 278 and 286 are represented in the table which is arranged so that gaps g=2 (mod 6), g=4 (mod 6) and g=O (mod 6) are found in separate columns. Gaps g=O (mod 6) occur much more frequently than the other two.

This is illustrated in figure 3 which also shows that In f as a function of g has a near linear behaviour for all three types. The "wild" behaviour fot gaps>250 would certainly correct itself if the range of primes in the study were extended. The area

13

below the curves for go;:2 (mod 6) fu'ld g=4 (mod 6) are equal as \\ill be sho\\TI shon!y.

The curve for g=2 (mod 6) behaves very well while the one for g=4 (mod 6) shows an interesting ripple etTect. In particular it shows a "bump" for g=70 \vhich showed up already in the smallest sample N<10 6 for which g=70 first appeared. Vvllat causes this high frequency for g=70" For a prire.e number p25 we have p=:::l (mod 6). Let q and p be two consecutive prilllcs forming the gap g=p-q. We distinguish between the follov.-ing cases: Shift

1.

q=l (mod 6) and p=1 (mod 6) => g=O (mod 6)

++

2.

q=l (mod 6) and p=-l (mod 6) => g=4 (mod 6)

+-

3.

q=-l (mod 6) and PEl (mod 6) => g=2 (mod 6)

-+

4. q=-l (mod 6) al1d p=-1 (mod 6) => g=O (mod 6) A seqi.le'ix cfconsecutive primes (\\-1m t.'le first prime =5) can be characterized by the

shift:::

+-+-+--H-++-++

-+-+

I II i I II I

+-+- .....

The longes! sequence of consecutive priInes = 1 (mod 6) tor r<109 is oflength 18:

450988159, 450988177, 4509gS207, 450988231, 450988241, 4509 88261, 450988297,450988333,450988339,450988381,450988387,450988399, 450988411,450988423,450988441,450988471, 450988477, 450988567

766319189, 7~6319201, 766319231, 7663l9237,766319249, 766319261, 766319273, 7563]9291,766319339, 766319357,766319363, 766319369, 766319423, 766319441, 766319453, 766319483,766319507, 766319549, ':J': '~95:3, 765319579,766319621,766319027 ~~~~ ~:lC :~~~;:~r

'-\-e

\\i;";.

c;: :???S=~7 路.1 c.[ G (illC.d 6) be f2, t~, ane f,: refeCtively for p<N. Then

~3~;e f2::":[4

if 'C-i;';

Lc.s.t ~[ift

is T - cJ.'1eT\\lse \\'~ \vill have f2=1'4+ i"

sL'pl", cL)~e: it "'F1:d 1:le reasonable to assu...-ne that, a: an arbitrfu"Y point in the of ~rjfts, t..~e rTvbabiLtie~ of tl:-i.ding the ne)..."1 shift to be -'--, ++, -+ or - are eql.-.aL i.e. ::' we :iefine r==f)(t'c+f:+L), r4=fJ(fo+f2+f4) and Fo=fc/(fo-r-f2+t4) we would have r,=F 4=025 a.TJ.d Fc=O.) Fis is not the case.

:::-. [0

~~~env2

14

Before looking into this let's fIrst consider a related question: Do primes=] (mod 6) (notation f~) occur with the same frequency as primes (mod 6) (notation fJ ?

Table 4 shows a study of the number of primes f _and f + congruent to -1 respectively 1 (mod 6) for primes less than Ht for k=l, 2,3, ... 9.

=-]

Within the range of this study we have L > is decreasing. Will eventually f _< f _ ?

f~.

However, the ratio r =(f _- f+)I(f _- f+)

We have proved that f2 = f4 (assuming the last shift to be +-). We will now study the relative frequencies defmed through

Again we have F2 = F4 and of course Fo=l - 2路h To study how F2 varies as we increase the number of consecutive primes p<lD" the execution of the program for gap statistics was stopped for k=l, 2, ... 9 to produce the data shown in table 5.

Table 4. Number of primes == -1 respectively 1 (mod 6). r=(f- - f.)/(f- + f.)

k L f.

L-f. f. +f .. r路l()l

2 1 1 0 2 0

3 85 81 4 166 241

12 11 1 23 435

4 616 611 5 1227 41

5 4805 4785 20 9590 21

6 39264 39232 32 78496 4

7

8

332383 332194 189 664577 3

2880936

9 25424819 25422713 2106 50847532 0.4

2880517 419 5761453 0.7

Table 5. Prime number gap distribution (mod 6) for primes <1()k k

1

2 3 4 5 6 7 8 9

g=2(mod6)

g=4(mod6)

g=O(mod6)

Total

i=2

2 9

1 8 56 378 2868 22837 189284 1616470 14107249

0 7 53 471 3853 32821 286C09 2528513 22633034

3 24 167 1228 9591 78497 664578 5761454 50847533

0.5 0.354166667 0.341317365 0.308224756 0.299134605 0.290941055 0.284819088 0.280566416 0.277442162

58

379 2870 22839

189285 1616471 14107250

15

Table 3.

9'1'2 2 6388042 8 5069051 14 -4690561 20 3528810 26 2190452 32 1359889 38 1103677 796213 50 678359 56 «-4581 62 247659 68 191321 7-4 137620 80 119756 6317-4 86 92 ....723 98 379-46 10-4 2-4215 110 2189-4 116 11385 122 7-408 128 5181 13-4 3881 1040 3970 1-46 1776 152 1288 158 886 16-4 661 170 607 176 332 182 238 12-4 188 194 109 91 200

....

206

212 In

224 230 236

2-42 2-48 254

260 266 272 284 290

....

35 21 18 17 10 8 6 3 3 1 1 2 1

P

3 89 113 887 2477 5591 30593 15683 31907 82073 3-4061 13-4513 -404597 5-42603 155921 927869 600W73

1388-483 1468277 58-45193 3117299 3851<459 6958667 7621259 6OJ.4247 8-421251 49269581 2028.5099 27915737 3f!D!fnJ7 36271601

13-4065829 166726367 378043979 232423823 2159-490407 327966101 409866323

607010093 216668603 367876529 191912783 1202....2089 9.... 192807 1438779821 1851255191 1667186<459 19488191~

gs.4

-4 10 16 22 28 3-4 040 -46 52 58 6-4 70 76

f 6386967 6568071 3527160 303Q3.48

23869....

1-430231 1306406 687135 511183 383239 253846

27283-4 122523

82

85030

88 9-4 100 106 112 118 124 130 136 1-42 1-48 15-4 160 166 172 178 18-4 190 196 202

65612 40821 39504 20996 17316 10863 7521 7111

208

21-4 220

226 232 238

3380

2393 1966 1561 1012 553 -430

292 235 205

112 71 56 38 36 15 3 8

P

7 139 1831 1129 2971 1327 19333 81-463 19609 ....293 89689 173359 212701 265621 5-4-4279 1100977 396733 10988-47 370261 13-49533 6752623 5518687 63710401 103-43761 2010733 4652353 33803689

83751121 3839-4127 39389989 79167733 1-4241-4669 7W16393

10753-4587 192983851 253878403

-47326693 519653371 525436489 673919143

2....

5

693103639

2SO

8 1 1 1 1 2 1

387096133

256 262 268 27-4 280 ~

16

1872851947 16-49328997 1579306789 1282-463269 1855047163 1-4531681-41

g.!)

6 110407651 12 8-472823 18 6-427670 2-4 -4600962 30 -4298663 36 23-41569 42 1904089-4 -48 119Q3.42 5-4 856601 60 789454 66 -466901 277514 72 78 233230 8-4 176328 90 133019 96 7186-4 102 52752 108 36484 114 26-413 120 23526 126 1-4-443 132 897-4 138 6567 1....

-410-4 150 04022 156 2152 1413 162 1271 168 174 729 180 638 3-42 186 192 219 221 198 129 20-4 141 210 216 50 31 222 21 228 23 23-4 15 2040 7 2-46 8 252 258 2 270 2 1 276 3 282 2 288

P

23 199 523 1669 -4297 9551 161-41 28229

35617 -43331 162143 31397 188029 -461717 -404851 360653

144-43J9 2238823

-492113 1895359 1671781 1357201 3826019

11981443 13626257 17983717 39175217 37305713 52721113 17051707 1-47684137 12345-4691 46006769 112098817 20831323 202551667 12216-4747 895858039 189695659 391995431 555142061 630045137 1316355323 13910-48047 6-49580171 -436273009 129-4268491

Rgure 3.ln[f) as a function of g for N<2路 10". g=O [mod 6) upper sofid fine, g=2 [mod 6) lower solid line and g=4 [mod 6) dashed line.

17

1

o.7

'"'0

.~

;--.

o.

"Vi o

1

,

I

i

,,

I

I

2

3

4

5

6

7

8

9

k Rgure 4. Relative frequency of prime gaps. F2 = F~ (thick line) and Fa (thin line).

Conclusion: F, <0.45 and F2 = F. >0.27 for primes <10'.

The approach to the values 0.5 and 0.25 that one would have expected is very slow and is slowed down with increasing Ie, - one realizes that the interval S<k<9 is ten times as large as the interval O<k<S. The data used is cumulative but even if we consider only the interval between 108 and 109 we have F2=12,659,767/45,6S0,669= 0.2771 compared to F2=O.2774 for the whole interval between 0 and 109 .

Question: Given an arbitrarily small number 8>0, does a prime PI exist so that F~.25+S for all P>PI?

IS

Chapter II On Smarandache Functions 1. Smarandache - Fibonacci Triplets We recall the defInition of the Smarandache Function Sen):

S(n) = the smallest positive integer such that S(n)! is divisible by n. and the Fibonacci recurrence formula:

Fa = F.I + F.} (n~) which for Fa = Fl = 1 defmes the Fibonacci series.

We will concern ourselves with isolated occurrences of triplets n, n-l, n-2 for which S(n)=S(n-l)+S(n-2) and pose the questions: Are there infInitely many such triplets? Is there a method of fInding such triplets which would indicate that there are in fact infInitely many of them?

A straight forward search by applying the defInition of the Smarandache Function to consecutive integers was used to identify the fIrst eleven triplets [1] which are listed in table 1. As often in empirical nwnber theory this merely scratched the surface of the ocean of integers. As can be seen from fIgure 1 the next triplet may occur for a value of n so large that a sequential search may be impractical and will not make us much \\<1ser.

# 1 2 3 4 5 6 7 8 9 10 11

Table 1. The first 11 Smarandache-Rbonacci triE2lets n SIn-I} Slnl 11 2·11 43 181 223 173 233 313 269 1109 569

11 121 4902 26245 32112 64010 368140 415664 2091206 2519648 4573053

19

5 5

29 18 197 2·23 2·41 2·73 2·101 2·101 2·53

Sfn-~

2·3 17 2·7 163 2·13 127 151 167 67 907 463

However, an interesting observation can be made from the triplets already found.

Apart from n=26245 the Smarandache-Fibonacci Triplets have in common that one member is two times a prime number while the other two members are prime numbers. This observation leads to a method to search for Smarandache Fibonacci triplets in which the following two theorems playa role:

I.

Ifn = ab with (a,b) = 1 and SCa) < SCb) then SCn) = SCb).

II.

If n = p' where p is a prime and a:::; p then SCp') = p.

~

â&#x20AC;˘

45CXXXX)

,

4XXXXX> 35CXXXX)

:DXXXX:l 25CXlXX)

: ,

2CXD.XD

.

;

15CXXXXJ 100XXX> 5CXXXXJ

1

0 0

2

1

....

6

4

Rgure 1. The values ofn for whidt the fIrst 11

8

12

10

Smarandadt~Fibonacci triplets

occur.

The search for Smarandache-Fibonacci triplets will be restricted to integers which meet the following requirements: n = xp' with a:::;p and SCx)<ap

(1)

n-l = yqb with lK-q and S(y)<bq

(2)

n-2 = zre with c:::;r and SCz)<cr

(3)

20

Table 2a. Smarandache-Flbonacci triplets #

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33

34 35 36

N 4 11 121 4902 32112 64010 368140 415664 2091206 2519648 4573053 7783364 79269727 136193976 321022289 445810543 559199345 670994143 836250239 893950202 937203749 1041478032 1148788154 1305978672 1834527185 2390706171 2502250627 3969415464 3970638169 4652535626 6079276799 6493607750 6964546435 11329931930 11695098243 11777879792

SIN) 4 11 22 43 223 173 233 313 269 1109 569 2591 2861

* *

3433

7589 1714 11?9 6491 9859 2213 10501 2647 2467 5653 3671 6661 2861 5801 2066 3506 3394 3049 2161 3023 12821 2174

*

SfN-l) 3 5 5 ?9 197 46 82 167 202 202 106 202 2719 554 178 761 662 838

* *

*

21

482 2062 10223 1286 746 1514 634 2642 2578 1198 643 3307 2837 1262 1814 2026 1?94 1597

* * * * * * * * * * * * * * * * * * *

* * *

SfN-2) 2 6 17 14 26 127 151 146 67 907 463 2389 142 2879 7411 953 467 5653 9377 151 278 1361 1721 4139 3037 4019 283 4603 1423 199 557 1787 347 997 11527 577

* *

*

0 0 0 -4 -1 -1 -1 -8

*

-1 0 -3 0 10 -1 5 -1 -5 -1 1 0 -9 -1 3 0 -5 0 -1 -2 -6 0 -1 5 -4 -4 2 6

Table 2b. Smarandache-Rbonacci triplets

# 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

N

StN)

13429326313 13849559620 14298230970 14988125477 17560225226 18704681856 23283250475 25184038673 29795026777 69481145903 107456166733 107722646054 122311664350 126460024832 155205225351 196209376292 210621762776 211939749997 344645609138 484400122414 533671822944 620317662021 703403257356 859525157632 898606860813 972733721905 1185892343342 1225392079121 129 4530625810 1517767218627 1905302845042 2679220490034 3043063820555 6098616817142 6505091986039 13666465868293

4778

S(N-l}

*

6883

2038 3209 4241 3046 4562 5582 11278 6301 10562 8222 20626 6917 8317 7246 6914 16774 7226 16811 21089 21929 13147 14158 19973 10267 18251 12202 17614 11617 22079 11402 14951 24767 31729 28099

*

* * * * * * *

* * * *

*

* *

*

22

1597 2474 1847 2986 3118 1823 463 1951 8819 3722 6043 6673 10463 2578 4034 3257 1567 11273 2803 12658 18118 20302 10874 3557 13402 10214 12022 9293 5807 8318 21478 7459 12202 20206 19862 16442

S(N-2}

* * *

*

* *

* * * * * * *

3181 4409 191 223 1123 1223 4099 3631 2459 2579 4519 1549 10163 4339 4283 3989 5347 5501 4423 4153 2971 1627 2273 10601 6571 53 6229 2909

* * * * * *

11807 3299 601 3943 2749 4561 11867 11657

7 2 -2 4 -10 -2 0 3 -1 -1 0 11 -5 -5 11 0 9 -1 0 0 -2 -5 -4 -2 -4 -3 -8

-1 11 5 2 2 7

p,q and r are primes. We then have S(n)=ap,S(n-l)=bq and S(n-2)=cr. From this and by subtracting (2) from (l) and (3) from (2) we get ap = bq + cr

(4) (5)

(6) The greatest common divisor (p.,qb)=l obviously divides the right hand side of (5).

This is the condition for (5) to have infinitely many solutions for each solution (p,q) to (4). Such solutions are found using Euclid's algorithm and can be \\TItten in the form: x= Xc+q\

y = yo - p"t

(5')

where t is an integer and (Xc, Yo) is the principal solution.

Solutions to (5') are substituted in (6') in order to obtain integer solutions for z. z = (yq b _ 1)Ire

Solutions were generated for (a,b,c)=(2,1,1), (a,b,c)=(l,2,1) and (a,b,c)=(l,l,2) \\ith the parameter t restricted to the interval -ll~tsl1. The result is sho\\TI in table 2.

Since the correctness of these calculations are easily verified from factorizations of S(n), S(n-l) and S(n-2) these are given in table 3 for two large solutions taken from an extension of table 2.

Table 3. Factorization of two Smarandache-Rbonacci triplets n= n-1 = n-2 =

16,738,688,950,356=22·3·31·193· 15,2692 16,738,688,950,355=5·197·1,399·1,741·6,977 16,738,688,950,354=2· 72 . 19·23·53·313· 23,561

S(n)= S(n-1 )= S(n-2)=

2·15,269 6,977 23,561

n= n-1 n-2

19,448,047,080,036=22.32.432.17,0932 19,448,047,080,035=5·7·19·37·61·761·17,027 19,448,047,080,034=2·97· 1,609·3,631· 17, 159

S(n)= S(n-1)= S(n-2)=

2·17,093 17,027 17,159

= =

Conjecture:

There are infinitely many triplets n, n-l, n-2 such that S(n)=S(n-l)+S(n-2).

23

Questions:

1.

It is interesting to note that there are <m1y 7 cases in table 2 where S(n-2) is two times a prime number and that they all occur for relatively small values of n.

V.rhich is the next case?

2. The solution for n=26245 stands out as a very interesting one.

Is it a unique case or is it a member of a family of Smarandache-Fibonacci triplets different from those studied here?

References: C. Ashbacher and M. Mudge, Personal Compllter World, October 1995, page 302.

2. Radu's Problem For a positive integer n, the Smarandache fimction S(n) is defmed as the smallest positive integer such that S(n)! is divisible by n. Radu [II noticed that for nearly all values ofn up to 4800 there is always at least one prime number between S(n) and S(n+l) including possibly S(n) and S(n+l). The exceptions are n=224 for which S(n)=8 and S(n+l)=lO and n=2057 for which S(n)=22 and S(n+l)=2I. Radu conjectured that, except for a finite set of numbers, there exists at least one prime number between S(n) and S(n+l). The conjecture does not bold if there are infinitely many solutions to the following problem.

Find consecutive integers n and n~ 1 for which two consecutive primes Pi and P~I exist so that Pi < Min(S(n),S(n-l); and Pi-I> .HaxfS(n),S(n-l)).

Consider

n+1 = xp,-s

(I)

and (2)

where Pr and Pr--l are consecutive prime numbers. Subtract (2) from (1). xp,-S _ ~'Pr-/ = 1

(3)

24

The greatest common divisor (PrS,p...J S) = I divides the right hand side of (3) 'Witich is the condition for this diophantine equation to have infintely many solutions. We are interested in positive integer solutions (x,y) such that the following conditions are met.

S(n+ 1) = sp., i.e. S(x) < sp.

(4)

S(n) = SPr-l , i.e. S(y) <sp,-l

(5)

In addition we require that the interval sp.S < q < SPr-l s is prime free, i.e. that q is not a prime.

Euclid's algorithm is used to obtain principal solutions (Xo, Yo) to (3). The general set of solutions to (3) is given by x= Xo+p.../',

y

=yo - Pr't

(7)

with t an integer.

These algorithms were implemented for different values of the parameters d=p.-l-Pr, s and t. The result was a very large munber of solutions. Table 4 shows the 20 smallest (in respect of n) solutions frn.md There is no indication that the set would be finite.

One pair of primes may produce several solutions.

Within the limits set by the design of the program the largest prime difference for which a solution was found was d=42 and the largest exponent 'Witich produced solutions was s=4. Some nmnerically large examples illustrating these facts are given in table 5.

To see the relation between these large munbers and the corresponding values of the Smarandache function in table 5 the factorizations of these numbers are given below: 1182293664715229578483018 = 2·3-89·193-431·16127812 1182293664715229578483017 =509·3253·16128232 11157906497858100263738683634 = 2.7.372.56671-553333 11157906497858100263738683635 =3-5.11.192.16433.553373 17549865213221162413502236227 =3-1 ]2·307·12671·553333 17549865213221162413502236226 =2.23-37.71.419.743-553373 270329975921205253634707051822848570391314 =2·33·47·1289·2017·119983-167441' 270329975921205253634707051822848570391313 = 37·231 17·24517·38303-167443"

25

Table 4. The 20 smaHest solutions which occurred for s=2 and d=2 #

n

Sin)

1 2 3 4 5 6 7 8

265225 843637 65303S5 24652435 35558770 40201975 45388758 46297822 67697937 138852445 157906534 171531580 299441785 551787925 1223918824 1276553470 1655870629 1853717287 1994004499 2256222280

9 10

11 12 13 14 15 16 17 18 19 20

Sjn+l}

PI

P2

206

202

302 122 926 1046 142 122 1142 214 1646 1718 1766 2126 2606 3398 3446 3758 3902

298 118

199 293 113 919 1039 139 113 1129 211 1637 1709 1759 2113 2593 3391

211 307 127 9':8 1049 149 127 1151

922

1042 146 118 1138 218 ~642

1714 1762 2122 2602 3394 3442 3754 3898

3994 4162

3998

4166

3433

3739 3889 3989

4159

0 0 -1

0 0

1 -4

0

223

0

1657 1721 1777 2129

0 0 0 0

2609

0

3407 3449 3761 3907 4001 4177

0 0 0 0 0 0

Table 5. Four numerically large solutions

Pairs of consecutive integers

SI)

d

5

1182':836647152':8578483018 11822936647152':8578483017 11157906497858100263738683634 11157906497858100263738683635 17549865213221162413502236227 17549865213221162413502236226 270329975921205253634707051822848570391314 270329975921205253634707051822848570391313

3225562 3225646 165999 166011 165999 166011 669764 669772

42

2

-2

4

3

0

4

3

-1

55333

0

55337 167441 167443

2

4

Pr. ~t

1612781 1612823 55333 55337

It is also interesting to see which are the nearest smaller Pi: and nearest bigger Pk-l primes to Sl = Min(S(n),S(n+IÂť and S2 = Max(S(n),S(n+IÂť respectively. This is sho\\-n in table 6 for the above examples.

26

PI:+1

3225539 165983 669763

3225562 165999 669764

3225646 166011 669772

3225647 166013 669787

108

30

24

Conclusion: There are infinitely many intervals {Min(S(n),S(n+l)),Max(S(n),S(n+l))} which are prime free.

References: I.M. Radu, Mathematical Spectrnm, Sheffield University, UK, Vol. 27, No.2, 199415, p.43.

3. The Smarandache Ceil Function Definition: For a positive integer n the Smarandache ceil function of order k is defmed through 1 S..(n) = m where m is the smallest positive integer for which n divides mil.

In the study of this function we will make frequent use of the ceil function defmed as follows:

I xl = the smallest integer not less than x.

The following properties follow directly from the above definitions:

1.

2.

3.

Sl(n) = n SJ«pCl) =

p

",'I;C

for any prime number p.

For distinct primes p, q, ... r we have SJ«pClq~ ... rO) =

i

a.'k-

-....(M:-

q~/k

Theorem L SJ«n) is a multiplicative function.

1 This fimdion has a great resemblance to the Smarandadle fimdion. h's definition was proposed by Ie. Kashihara (Japan) and conveyed to the author by R. ~uller.

27

A fimction fen) is said to be multiplicative if for (nl,TI2) = 1 if it is true that f(n]fl2) = f(nl)f(n2). In our case it follows directly from (3) that if (nl,n2) = 1 then St;(n]fl2) = Slc(nl)Slc(n2).

However, consider n=nln2 when (nl,TI2);tol. In a simple cast: let nJ=ml·p(J. and n2=m2·ptl with (m],m2)=1 we then have Slc(n) = Slc(ml)Slc(m2)-P ((J.~Il)lk which differs from S~nl)S~n2) whenever r(a+~)Ik1 r aIk1 + r~/k1. In fact one easily proves that r (a+~)Ik1= r aIk1+ r ~/k1or r (a+~)Ik1= r aIk1+ r ~/k1- 1.

'*'

Express n in prime factor form n=p(J.qll···ro and apply (3). We then see that all prime powers in Sk-I(n) are less than or equal to those ofSlc(n), i.e. Shl(n) Slc(n).

I

m

Theorem For sufficiently large values ofk we have Slc(n)=I1p; where the product is taken over all distinct primes p; of n.

By extending the argument in theorem IT we have that, if j=max(a, Slc(n)=pq···r for k ~ j.

~,

.,. 8) then

Corollary 1. Slc(p) = p for any prime number p.

Corollary 2. Ifn is square free then S2(n) = n.

Theorem IV. k exists so that

~n!)

=

P#. where p is the largest prime dividing n.

p# denotes the product of all primes less than or equal to p. Let's write n! in prime factor form. n!=2(J.3 tl ... pY, where a>~> ....>r. In order to apply theorem ill we need to fmd a. Consider 1·2·3·4·5·6 ... n. This product contains [n!l2] even integers, [n!/4] multiples of 4, etc .... and finally [n!l21 multiples of 2°, where 2°!ill!<2~1. 8 is determined by 8=[log nIlog 2]. From this we fmd that S~n!) = p# for t5

k=a

= ~)n!/ 2r] r=!

28

Table 7. The Smarandache ceil function

n

S3

S2 <I

8 9 12 16 18 20 2<1 25 27 28 32 36 40

<14 45 48

<19 50

52 54 56

60 63

64 68 72

75 76

S4

<I

<I

6 10 12 5 9 1<1 8 6 20 22 15 12 7 10 26 18 28 30 21 8 34 12 15

4

3 <I

12

<I

<I

4

2

200 204

207

38

26 6

28

28

~~

6

6

10 52 18

112 116 117 120 121 12<1 125 126

2

6 14

100 104 lOB

4

10

33

90

2

6

99

84 88

n 135 136 140 144 147 148 150 152 153 156 160 162 164 168 169 171 172 175 176 180 184 188 189 192 196 198

2

92 96 98

81

S7

56

3 6

20 9 <12 44 30 46 24 14

80

S5

2

20 9

10 3

208

212 216

22

220

22<1 225 12

12

6

228

14

58 39

60 11 62 25 <12

30

~~

8

5 4

4

4

2

29

232 234 236 240 2<12 243 244 245 248 250 252 256 260 261 264

S2 45 68 70 12 21 74 30 76 51 78 40 18 82 84 13 57

S3

S5

S4

sa

S7

S6

15 34 12

6

38

20 18

20 6

10

<12

86

35 44 30 92 9<1 63

24 14 66 20 102 69 52 106 36 110 56 15 114 116 78 118 60 22 27 122 35 124 50 <12 16 130 87 132

44 1 <16 21 12

22

12

12

10 52

26

6 28

28

14

58

60

30

9

9

3

4

<I

62 10 8 66

4

<I

2

Calculations.. Calculation of ~ n) for n < I ()()() were carried out in Ubasic, which has a built in ceil function. The result is shown in table 7. Since Sin) = n for square free numbers these have been excluded from the table. When Stin) is square free the entries for larger values of k become repetitive. Instead of repeating these values the corresponding spaces in the table have been left blank.

4. The Smanmdache Pseudo Function Z(n) Defmition2: Z(n) is the smallest positive integer m such that 1+2+ ... +m is divisible byn Alternative formulation: For a given positive integer n, Z(n) equals the smallest positive integer m such that m(m+ l)12n is an integer.

The following properties follow directly from the definition:

1.

2.

3.

4.

5.

Z(1)=1 Z(2)=3

For any odd prime number p, Z(p)=}r 1

By extension of (3) we have Z(IfW-I In the special case n=~ we have Z(2k)=2k-l_1

Calculation of Z(n) We need to find m so that m(m+l)=2nk has a positive integer solution for the smallest possible positive value ofk.

m=

-1+Jl+8kn 2

For a given value ofn the smallest square 1+8kn is found by executing the following program lines in Ubasic where effective use of the ISQRT(x) has been made:

1 Definition by KKasIOOara (Japan) OOIIveyedtotheaUlhor by R Muller, Emus Univasay PraI,

USA 30

10 INPlIT "n ";n 20k=O 30 inc k 40 x=1+8*k*n 50 ifx<>(isqrt(xÂť"2 then goto 30 etc - to evaluate m The complete program has been implemented for table 8.

~IOOO.

The result is displayed in

Theorem: If n=pq, where p and q are two distinct primes with g=q-p, then Z(n)=Min(p(qk+lyg where pk+I=O (mod g), q(pk-Iyg where pk-l=O (mod gÂť

Proof: We have to consider three cases:

1.

2.

3.

p / m and q / (m+ I) which, since we assume q>p, we distinguish from p/(m+l)andq/m pq/ (m+l)

Case I. Consider px=m and qy=m+ I which together with g=q-p gives

(I)

p(x-y)=gy-I

Since we must have p / (gy-I) we can put gy-I =pk where k / (x-y). Our solution for y then becomes y=(pk+ I yg with pk+ I =0 mod g)

(2)

Inserting this in (1 ) results in p(x-(pk+lyg)=pk from which

(3)

x=(qk+lyg which we insert in m=px to obtain m=p(qk+lyg where k is determined through pk+I=O (mod g)

31

Table 80. Z(n) for n:s; 1000, n non-prime

1 2 4 6 8 9 10 12 14 15 16 18 20 21 22 24 25 26 27 28

30 32 33 34 35 36 38 39 40 42 44 45 46 48 49 50 51 52 54 55 56 57

Z(n) 1 3 7 3 15 8 4 8 7 5 31 8 15 6 11 15 24 12 26 7 15 63 11 16 14 8 19 12 15 20 32 9 23 32 48 24 17 39

27 10 48 18

n 58 60

62 63 64 65 66 68 69 70 72 74 75 76 77 78 80 81 82 84

85 86 87 88 90 91 92 93 94 95 96 98 99 100 102 104 105 106 108 110 111 112

Z(n) 28 15 31 27 127 25 11 16 23 20 63 36

24 56 21 12 64 80 40 48 34 43 29 32 35 13 23 30 47 19 63 48 44 24 51 64 14 52 80 44 36 63

n 114 115 116 117 118 119 120 121 122 123 124 125 126 128 129 130 132 133 134 135 136 138 140 141 142 143 144 145 146 147 148 150 152 153 154 155 156 158 159 160 161 162

Z(n) 56 45 87 26 59 34 15 120 60 41 31 124 27 255 42

39 32 56 67 54 16 23 55 47 71 65 63 29 72 48 111 24 95 17 55 30 39 79 53 64 69 80

32

n 164 165 166 168 169 170 171 172 174 175 176 177 178 180 182 183 184 185 186 187 188 189 190 192 194 195 196 198 200

201 202 203 204 205 206 207 208

209

210 212 213 214

Z(n) 40 44 83 48 168 84

18 128 87 49 32 59 88 80 91 60 160 74 92 33 47 27 19 128 96 39 48 44 175 66 100 28 119 40 103 45 64 76 20 159 71 107

n 215 216 217 218 219 220 221 222 224 225 226 228 230 231 232 234 235 236 237 238 240 242 243 244 245 246 247 248 249 250 252 253 254 255 256 258 259 260 261 262 264 265

Z(n) 85 80 62 108 72 55 51 36 63 99

112 56 115 21 144 116 94 176 78 84 95 120 242 183 49 123 38 31 83 124 63 22 127 50 511 128 111 39 116 131 32 105

Table 8b. lin) for n s; 1000, n non-prime n 266 267 268 270 272 273 274 275 276 278 279 280 282 284 285 286 287 288 289 290 291 292 294 295 296 297 298 299 300

301 302 303 304 305 306 308 309

310 312 314 315 316

lin) 56 89 200

80 255 77 136 99 23 139 62 160 47 71 75 143 41 63 288 115 96 72 48 59 111 54 148 91 24 42 151 101 95 60 135 55 102 124 143 156 35 79

n 318 319 320 321 322 323 324 325 326 327 328 329 330

332 333 334

335 336 338 339 340 341 342 343 344 345

346 348 350 351 352 354 355 356 357 358 360 361 362 363 364 365

lin) 159 87 255 107 91 152 80 25 163 108 287 140 44 248 36 167 134 63 168 113 119 154 152 342 128 45 172 87 175 26 319 59 70 88 84 179 80 360 180 120 104 145

n 366 368 369 370 371 372 374 375 376 377 378

lin) 60 160 81 184 105 216 187 125 47 116 27 95 126 191 255 55 192 171 96 39 68 48 131 196 79 143 199 56 224

380

381 382 384 385

386 387 388

390 391 392 393 394 395 396 398 399 400 402 403 404 405 406 407

200

155 303 80 28 110 255 40 137 103 118 207 165

408

410 411 412 413 414 415

33

n 416 417 418 420 422 423 424 425 426 427 428 429 430 432 434

446 447

lin) 64 138 76 104 211 188 159 50 71 182 320 65 215 351 216 29 327 114 72 175 98 51 111 89 223 149

448

384

450 451 452 453 454 455 456 458 459 460 462 464 465 466 468

99 164 112 150 227 90 95 228 135 160 132 319 30 232 143

435

436 437 438

440 441 442 444 445

n 469 470 471 472 473 474 475 476 477 478 480

481 482 483

484 485

486 488

489 490 492 493 494 495 496 497 498 500 501 502 504 505 506 507 508 510 511 512 513 514 515 516

lin) 133 140 156 176 43 236 75 119 53 239 255 221 240 69 120 194 243 304 162 195 287 203 208

44 31 70 83

375 167 251 63 100 252 168 127 84 146 1023 189 256 205 128

Table Be. lfn) for n

~

1COO. n non-prime

n

lfn)

n

lIn)

n

lfn)

n

lIn)

n

517 518 519 520 522 524 525 526 527 528 5'Zt

187 111 173 64 116 392 125 263 186 32 528 159 117

565 566

225 283

175

667

308

668

115 167

567

161 496 75 143 191 287 275 512

616 618 620 621 622 623 624 625 626 627 628 6'Zt

669 670 671 672 674 675 676 678 679

580

Sf,

581

279 161 311 266 351 624 312 132 471 221 35 79 210 316 254 159 195 87 71

715 716 717 718 720 721 722 723 724 725 726 728 7'Zt

530

531 532 533 534

535 536

537

246 267 214 335

543

179 268 98 80 271 180

544

255

538

539 540

542

s.s 546 548

549 550

551 552 553 554 555 556 558

559 560

561

~

109 104 136 243 99 57 207 237 276 74 416 216 1'Zt 160 33 280

47

568

570 572 573 574 575 576 578 579

5s2 583 584 585 586 588 589

590 591 592 594 595 596 597 598

288

192 144 83 96 264 511 90

630 632

633 634

635 636

292

637

48 247 59 197

638 639 640

642

480

644

600

'Zt6 34 447 198 91 224

602

300

603

134 151 120

654 655

512 174 60

658

662

234

663

135

664

604

605 606 608 609

610 611 612 614 615

645

646 648

649 650

651 652

255

107 160 1'Zt 152 80 176

m

62 488

~7

665

108 130 287 72 140 120 331 51 415 189

164

666

36

m

656 657 660

.34

680

222 200

121 63 336

324 168 339

97 255

681

227

682 684 685 686

340

7~

152 274

731 732 734 735 736 737 738 740 741 742 744 745 746 747 748 749 750 752 753 754 755 756 758 759 760 762 763 764

687 688 689

690 692 693 694 695 696 697 698 699 700 702 703 704 705 706 707 708 710 711 712 713 714

343 228

128 52 275 519 98

347 139 144 204

348 233

175 324 37 384

140 352 202

176 284

315 623 92 84

lIn)

65 536 239

359 224 308 360

240 543

174 120 272 728 219 85

183 367 195 575 66 287 184 38 371 464

149 372 332

407 321 375 704 251 116 150 216 379 230

95 380

217 191

Table ad. lin) for n ~ 1000, n non-prime n 765 766 767 768 770 771 772 774 775 776 777 778 779 780 781 782 783 784 785 786 788 789 790 791 792 793 794 795 796 798 799 800

801 802 803 804 805 806 807 808

810 812

lin)

135 383

117 512 55

257 192 171 124 96 111 388

246 39 142 68 377 735 314 131 591 263 79 112 143 182 396 105 199 56 187 575 89 400

219 200

69 155 269 303 80 231

n 813 814 815 816 817 818 819 820 822 824 825 826 828

Z(n) 270 296 325 255 171 408

834

90 40 411 720 99 412 207 415 276 767 391 416

835

334

836 837 838

208

830

831 832 833

840

841 842 843

844 845 846 847 848

849 850 851 852 854 855 856 858 860 861 862

216 419 224 840 420 281 632 169 188 363 159 282 424 184 71 244 170 320 143 215 41 431

n 864 865 866 867

lin)

868

216 395 144 402 544 387 436 125 72 439 293 319 440 272 59

512 345

432 288

869 870 871 872 873 874 875 876 878 879 880

882 884 885 886

443

111 126 355 242 223 94 447 179 511 207 448 434 224 424 164 42 112 180 452 680 404 104

888

889 890 891 892 893 894 895 896 897 898 899 900

901 902 903 904 905 906 908 909 910

35

n 912 913 914 915 916 917 918 920 921 922 923 924 925 926 927 928 930 931 932 933 934 935 936 938 939 940 942 943 944 945 946 948 949 950 951 952 954 955 956 957 958 959

lin)

95 165 456 60 687 392 135 160 306 460 142 231 74 463 206 319 155 342 232 311 467 220 143 335

312 375 156 368 767 189 43 552 364 75 317 272 423 190 239 87 479 273

n 960 961 962 963 964 965 966 968 969 970 972 973 974 975 976 978 979 980 981 982 984 985 986 987 988 989 990 992 993 994 995 996 998 999 1000

lin)

255 960 259 107 240 385

252 847 152 484

728 139 487

m

671 488

88 440

108 491 287 394 203

140 208

344

44 960 330

496 199 248 499 296 624

Case 2. Consider px=m+ I and qy=m together with g=q-p. Repeating the calculation in case I results in: m=q(pk-l)/g where k is determined through pk-l=O (mod g)

Case 3. Consider m=IXJ.-l. We must have IXJ.-I<p(qk+l)/g to beat case l. This inequality can be written p(q(g-k)-l)<g where g>k and q22 from which we see that this inequality is impossible. The argument is similar to reject case 3 in comparison with case 2. m is therefore given by the minimum value for m as calculated in cases I and 2, i.e.

Z(n)=Min(p(qk+l)/g where pk+l=O (mod g), q(pk-l)/g where pk-l=O (mod g)) Corollary: Z(n) is not a multiplicative fimction.

The above theory has been used to calculate Z(IXJ.) for a few prime number gaps g=qp. The result is shown in table 9.

Table 90. lIn) for the first 10 prime gaps g=1O and g=30 Gap: q-p=1O p q 139 181 241 283

337 409 421 547 577 631

149 191 251 293 347 419 431 557 587 641

n 20711 34571 60491 82919 116939 171371 181451 304679 338699 404471

lIn) 2085 3438

6024 24904 35047 17178 18102 91348 101551 40383

Gap: q-p=30 p 4297 4831 5351 5749 6491 6917 7253 7759 7963 8389

36

q 4327 4861 5381 5779 6521 6947 7283 7789 7993 8419

n 18593119 23483491 28793731 33223471 42327811 48052399 52823599 60434851 63648259 70626991

lIn) 8056874 782621 10557522 12182131 15519980 11212457 22890468 22159704 14850994 25896843

Table 9b. lIn) for the first 10 prime gaps g=20 and g=40 Gap: q-p=20 p q 887 907 1637 1657 3089 3109 3433 3413 3947 3967 5717 5737 5903 5923 5987 6007 6803 6823 7649 7669

n 804509 2712509 9603701 11716829 15657749 32798429 34963469 35963909 46416869 58660181

lIn) 120631 949460 4321510 1757695 2348464 11479736 12236918 5394286 16245563 26396698

Gap: q-p=40 p q 19333 19373 20849 20809 22573 22613 25261 25301 33247 33287 38461 38501 45013 45053 48907 48947 52321 52361 60169 60209

37

n 374538209 433846841 510443249 639128561 1106692889 1480786961 2027970689 2393850929 2739579881 3622715321

lIn) 28090849 97615018 38283808 303586698 470345309 703374768 152098927 179537596 68488188 815109442

Chapter III

Loops and Invariants

In many practical as well as theoretical processes we repeat the same operation on an object again and again in order to arrive at a fInal result, or sustain a certain state or maybe simply to see what is going to happen. Each time a hammer hits a nail the nails sinks a bit deeper until it can sink no further. This repeated operation has resulted in an invariant state. A different situation occurs in an engine when energy is used to make a piston perform cycles or loops. Many similar phenomena occur when repeating processes on numbers through iterations. Before going into the problems of this section let us consider the iteration process itself.

Let l(n) defme an operation to be carried out on n. If we apply this operation to len) itself we say that we perform an iteration and could write this as l(l(nÂť. After a number of iterations we could have something like l(l( .. .l(n) ... Âť, which we ""ill write l(k)(n) to indicate the kth iteration. Alternatively we can use llk to denote the result of the kth iteration and no to denote the starting value. We then have nl = l(no), Let us apply this to a simple case where len) is defmed through l(n) = lin. If we take no=l then llk=l for all k, i.e. the result of the iteration is invariant. If we take no=2 then llk=1I2 for odd values ofk and llk=2 for even values ofk, i.e. we have an iteration resulting in a loop of length 2. If we apply the iteration process to l(n)=n 2 then the result will be forever increasing and we say that the iteration is divergent.

Since we are dealing with a very important concept which has attracted a lot attention some of the topics in this section have been dealt with before. In particular Perfect Digital Invariants [1], which has recently been reactivated under another name "Steinhaus' problem" [2]. However, all results presented here have been generated in recent studies carried out by the author and have been retained even though some of them may duplicate earlier results. This proved necessarY in order to arrive at a consistent presentation and some new results. 1. S. Madachy has made me aware of more literature on this interesting topic some of which is listed in the references.

38

1.

Perfect Digital Invariants and Related Loops

For an arbitrary positive integer

we defme Nk-l = l(Nk) through

For a given positive integer no the iteration process nl = l(no), n2 continued until one of the following situations is reached:

I.

I( ilk) = ilk

ll. l(nk-L) = nk ill. I(nk) ~ ex;

l(nl), ... is

Perfect Digital Invariant (PDI) Loop oflength L Divergent

Case III is impossible. It follows from the following determination of the upper search limit for each value of q. Let us consider the largest possible n-digit number N = 999 ... 99 = Hf-l. We have l(N) = n路9q and need to determine Nmax so that N-I(N禄O for all N>Nmax . Let u be the largest value ofn for which lOn-I-n路9 Q < 0, i.e.

then there is a smallest positive integer a, lsas9, such that

This gives Nmax = (a+ 1)- 1Oll -1 as an upper limit for solutions. Nmax could be improved by looking for a smaller value than 9 for the second most significant figure but this would give more complications than benefits in computer implementation. That Nmax exists proves that the iteration process does not diverge since after a number of iterations larger than Nmax a pre\"iously assumed value must be repeated completing a loop or collapsing on an invariant.

Only a small subset of all integers < Nmax needs to be used as input numbers in a search program. The follo'Wing two input criteria greatly reduce the computer execution time.

39

1. The order in \\'hich digits occur in an input number is of no importance. No

=

2337 will give the same result as No = 3732. A number whose digits are a pennutation of an already used input number will therefore be rejected. As an example take q=S for \\'hich Nmax = 299999. In this case an input number needs to have maximum 6 digits of \\'hich there can be at most 6 ones or twos and a maximum of S of the other digits.

2. Input criterion number 1 is

used so that the search always proceeds from a smaller to a larger input number. If for any input number No we have l(No)<No then the iteration process is aborted since l(No) has either been dealt with before or doesn't meet criterion number 1.

Complete solutions were calculated for q = 2,3,4, ." IS. Apart from the trivial case l(N)=1 these solutions are given in table 1 together with the upper search limit for each q. The longest loop is of length 381 for q = 14. There are no Perfect Digital Invariants (PDIs), i.e. solutions to N = l(N) for q=2, q=I2 and q=IS. References:

I. 2.

3.

Lionel E. Deime1 Jf. and Michael T. Jones. Journal of Recreational Mathematics, pgs 87-108. Vol. 14.2 Personal Computer World, page 333, January 1996 Dean Morrow,Joumal ofRecreational Mathematics, pgs 9-12, Vol. 27.1

40

Table 1a. PDls and related loops q

2 3

Nmax S=1;N

length of loop

Smallest

199/4 2999

8

4 153 370 371 407 136 919 55 160 1634 8208 9474 2178 1138 4150 4151 54748 92727 93084 194979 58618 89883 10933 8299 8294 9044 24584 9045 244 548834 63804 282595 93531 239459 17148

term

$=76 2 2 3 3 4

29999 $=153

1 2 7

5

299999 $=345 1 1 2 2 4 6 10 10 12 22 28

6

3999999 $=401

2 3 4 10

30

41

Largest term

N

145

3 17 16 19 2

244 1459 250 352

6514 13139

11 6

3 1 8 2 6 135

10

76438 157596 73318 150898 183635 133682 180515 167916 213040 313625 845130 650550 1083396 1758629

1 23 9 13 24 33

93 112 21 2 5 71 5 167 150

Table lb. PDls and related loops q

7

Nmax

length of

S=kN

loop

4路10'-1

1 1 1 1 1 2 2 3 6 12 14 21 27 30 56 92

S=1012

8

4路1 QB-l

S=1544

9

Smallest term 1741725 4210818 9800817 9926315 14459929 2755907 8139850 2767918 2191663 1152428 922428 982108 253074 376762 86874 80441 24678050 24678051 88593477 7973187 8616804 6822 146511208 472335975 534494836 912985153 144839908 277668893 180975193 409589079 52666768 20700388 62986925 180450907 42569390 52687474 322219 41179919 32972646 2274831

4.109 -1 S=5058

1 1 3 25 154 1 1 1 1 2 2 3 3 4 8 10 10 19 24 28 30 80

93

42

largest term

6586433 9057586 8807272 9646378 14349038 16417266 14600170 18575979 192100J3 19134192 14628971

77124902 149277123 153362052

1043820406 756738746 951385123 1339048071 574062013 1212975109 931168247 857521513 1001797253 708260569 1298734342 1202877221 1724515947 1430717631

N

2 7 10 12 1 1 30 46 21 70 28 93 141 225 114 210 19 188 12 14 828 482 34 22 53 34 45 3 8 13 171 545 164 87 403 373 262 1434 1133 273

Table le. PDls and related loops q

Nmax S=!: N

10

4.10 10-1 S=7408

11

4路10 1 Q

length of loop 1 2 6 7 17 81 123 1

S=7628

12

4.10 12..1 S=9466

1 1 2 3 3 3 5 5 7 10 18 20 42 48 117 118 181 5 40 94 133

Smallest

lerm 4679307774 304162700 123148627 1139785743 62681428 20818070 192215803 32164049650 32164049651 40028394225 42678290603 44708635679 49388550606 82693916578 94204591914 4370652168 2491335968 4517134494 6666140097 416528075 2181207047 9005758176 3967417642 12650989279 2075164239 195493746 101858747 739062760 872080538 8922100 98840282759 2700216437 4876998775 16068940818

43

largest term 344050075 7540618502 9131926726 13957953853 15434111 703 14230723551

N

13

159 1084 706 711 4733 30 63

5 6 7 11 21 11346057072 71768229638 33424168842 36704410767 103153306403 28167146357 71727610926 98110415227 128870085703 127554589656 106744983639 134844138593 169812860326 165906857819 176062673167 785119716404 1157645923834 1281243062759 1200615480 166

58 75 2 3 54 43 50

100 486

1205 1075 1015 1566 609

1142 128 1557 1883 5897

Table 1d. PDls and related loops

q

13

Nmax

length of

Smallest

largest

S=LN

loop

term

term

4.10 13-1 S=35177

14

4.10 14-] S=40180

15

4.10 15-1 S=48785

N

1

564240140138

3

3396705890823

6294418483143

5

2310198454262

10830899644190

50

5

3656948275943

6405584099531

1601

10 240

8

6882715L5816

5647840775906

8846

9

11 13928853354

5840462013812

503

13

672769417360

6842950405261

240

16

662218816395

10829615029038

18

100555910945

7735617503306

3403 404

22

221192011997

7722486519974

821

31

759039131807

12916172647637

625

100

98183216073

10719727658703

6794

146

111240729637

11270705761846

11753

1

28116440335967

4

6674458190799

74548238736415

9

4

27510477911590

47800729611562

428

60

5

841332967215

57268678554888

154

6

5911230616470

55918604360271

441

6

22955961974580

94220062144011

311

7

5833130055708

97267770775241

35

14

1447300158177

92876091448633

5119

15

4423275478678

78789825354783

1698

65

931899363208

118800319349172

3517

96

157270047611

92354596687594

9448

381

78924999008

142415666495594

18959

3

255349823145519

447090882837630

17

8

76058866219899

729415146625008

10157

11

70433029388274

657638038056753

518

12

41845353013296

632509424234352

1261

15

14745577655668

619115571288208

177

19

14746637048586

895875047683584

362

30

1096193499692

1030397118565900

941

46

10059036985836

662536011243090

1302 18270

75

593343453343

864468451964287

216

92683904599

1065173414568230

2400

362

595476555320

1145004602355100

13379

44

2.

The Squambling Function

The squambling function U(N) originates from The Penguin Dictionary of Curious and Interesting Numbers, page 169. In its present form the problem was formulated by G. Samson in the November 1995 issue of Personal Computer World: For a given integerN = ao+ adO + ... +anlO" > lOn, an;z: 0, the squambling function is given bv

Iterating the squambling function will result in an invariant or a loop. As in the previous problem there are no starting values for which the process diverges. To see this consider

If No exists for which U(N) < No for all N > No then the squambling function must return to one of its previous values after a sufficiently large number of iterations, i.e. the process converges (or goes into a loop). No exists and is estimated to be less than 10 43 from the inequality below

More detailed considerations may considerably lower this upper limit for No, but in any case the upper limit is so large that a complete search for invariants and loops is hardly feasible. The result of a search for squam-invariants and squam-loops for integers up to 106 is summarized in table 2, where L=the total number of numbers in the loop (=length of the loop, which equals 1 for an invariant), M=the smallest number in the loop, N=the largest number in the loop and Q=the number of loops found for initiating integers less than 10 6 .

Table 2. Squam-Ioops and squam-invariants L M N Q

1 1 1 165421

1 43 43

613722

1 63 63 4617

1 47016 47016 125

1 542186 542186 6

45

3 126529 4787463 13

8 579 59830 2!J77

105 5 43055027 214018

If there are more invariants and loops then they must have a smallest element greater than 106, or more precisely a smallest element "hich is larger than the upper limit for a previous search. This has been used in an extended search for starting integers up to 50,000,000. No new loops were found. Question: Are there any?

Let us fmally consider what happens if we reduce the powers to which each digit is raised by one. This makes the process a fortiori convergent as can be seen from the arguments used previously. The iteration process was examined for starting integers up to 10 5 No loops were found. Apart from iterations ending on one-digit invariants a number of other invariants were found.

Table 3. The number of start integers Q below lCJ5 resulting in an invariant I

Question: Are there no loops in this case?

3. Wondrous Numbers This study deals with the extended Wondrous Numbers Conjecture stated as follows

by B.C. Wiggin [1]:

Any integer n 2 (D-I) may be directed through a series of iterations as a function n "" R (mod D). Starting with no the series of iterated Wondrous Sumbers are nl, n2,.路路 Ilk, ...

Definition: Let 11k=fu (mod D) iffu=O then 1lk-1=1lk1D ifl~g)-2 then 1lk-1=DJc(D+I)-fu iffu=D-I then 1lk-1=llk(D+I}+1 It is conjectured that the series ultimately converges to n<D. This study will show that the above conjecture does not hold.

To examine the behaviour of the above process assume that for a given Ilk we have Ilk "" 0 (mod D)

46

in which case I1k~1

=

11kiD;

Let I1k~1 '" R!c+l (mod D)

We have to consider three cases

Case I R!c-l =0

then

Case II

then I1k+2 = I1k+ 1(D+ 1) - R!c+l

From this we see that I1k+2 '" 0 (mod D) and substituting from (l) that I1k+2

< I1k(1 + lID)

IfR!c+2i-l stays within the above limits for I=I, 2, ... m then I1k+2m <

I1k(1 + lID)

Case ill R!c+l =

D-l

As in case IT I1k+2..o (mod D) but

47

(4)

For the series to divC%ge the case R=O must occur with a frequency which is much higher than the expected lID (on the average). In a worst case scenario let us assume that R ~ 0 so many times that it balances the effect of the occurrence of case I. Ignoring (4) which can be considered as compensated by ignoring Rt.-J in (3) we then have. nc(1

+ IlDjlfY ;

Ilk

For D ; II this gives m "" 56 just to hold the balance. If remainders R are evenly distributed then m would vary around 10. It is very safe to say that the series does not diverge. However, it does not always converge to n < D as conjectured Brendan Woods (2] reported that she failed to get termination with (D,N) ; (13,70), (14,75), (58,59), (82,83) and (l98,199). The reason is that these cases produce loops as shown in table 4. No = 199 , D = 198 produces a very loog loop oflength 2279 requiring 2499 iterations.

Charles Ashbacber [3) investigated aU divisors in the interval 3 :s D :s 12 for aU initial values Do < 14000000. Even with the most effective programming and up to date equipment this is a very impressive piece of work. In all cases iterations terminated on a ooe-digit number. He listed those cases for D=II and D=12 v.bere more than 1000 iterations were required before the terminal value was reached and made the

following observation .....for cenain series of ascending n the mmrber of cycles descended in steps of 2 contrary to the expected behaviOllr that the mmrber of cycles rises with the size of the mmrber ". He adds that no justification bas been found for this and challenges readers to further explore this behaviour. The remainder of this section will be devoted to an explanation of this mystery though an analysis of the case D = 11. A similar analysis to the one below bas been carried out for D = 12 with similar results.

Since aU iterations result in a one-digit terminal value aU cases with

no <

14000000

which require more than 1000 iterations can be classified according to their terminal value. This is done in the column marked p in table 6, which contains Ashbacher's table as a subset There are only three different terminal values 3, 7 and 8.

Table 5 shows the ISO first iterations for Do ; 1345680, which is the first entry in Ashbacher's table. Table 5 confirms the above theory that at least every second R is zero. This explains the step 2 decrease with ascending n since this causes the terms to oscillate. In geoc:ra1 every second term was out of bounds of the investigation which was limited to no<l4000000. These cases with long cycles are due to loog "freak" 48

oscillate. In general every second term was out of bounds of the investigation which was limited to 110<14000000. These cases with long cycles are due to long "freak:" sequences if iterations with remainders R=O. Let's apply (3') to the fIrst occurrence of R=O for even k in table 5. nl = 16148154, nl67 = 167493499. n167< 16148154(1 + 11l1)821l21 '" 167494096 This, as is seen, is a very good approximation.

Table 4. Loops instead of anticipated convergence

no

n

13 14 58 82 198

70 75 59 83

199

loopstorb

First term of

loop

leng1hof

for :

loop

finishes for: 162 91 555 925 2499

loop

92 13 19 153 221

1911 166 4002 13120 61380

71 79 537 773 2279

We also see from table 5 that it contains a sequence of numbers which also occur in table 6, i.e. the initiating value 110 = 1345680 is the "grandparent" of a whole family of larger initiating values having the same terminal value 8. However, table 6 also contains initiating values with terminal value 8 which do not occur in table 5. At some stage, however, these will merge with the iteration process for the "grandparent" .

These cases have been identifIed and labeled in column c (c="child") in table 6. These "children" have been used as starting numbers for iteration processes to see at which point they will join the "grandparent" iteration. The result is shown in tables 7 and 8, where K-c is the number of iterations for the child (which may have a number of children of its own as can be seen from table 6), and K -p is the number of iterations for the grandparent before merging occurs. It is amazing how soon this happens. Only the 8-family child 11700624 makes it on its own almost to the end. N-max is the maximum value for the iteration process which occurs after K-m iterations. The 3family is childless.

References:

1.

2.

3

B.C. Wiggin, Journal ofRecreational Mathematics, pgs 52-56, Vol. 20.1 M. Mudge, Personal Computer World, page 335, Dec. 1995 C. AshbaÂŁher, Journal ofRecreational Mathematics, pgs. 12-15, Vol. 24.1

49

Table 5. The first 180 iterations for D=11, N=134568O

k

R

n

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

16148154 17616159 19217627 20964680 22870551 24949683 27217828 29692168 32391447 35336125 38548499 42052901 45875885 50046414 54596080 59559357 64973843 70880557 77324236 84353709 92022227 100387881 109514053 119469867 130330761 142179004 155104367 169204761 184587007 201367639 219673784 239644119 261429949 285196307 311123241 339407167 370262365 403922574 440642807 480701243 524401350 572074195 624080941 680815564 742707889

55 57 59 61 63 65 67 69 71 73 75 77 79 81

83 85 87

89

t

R

n

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44

9 1 4 9 9 8 8 9 10 1 7 7 6 8 3 1 10 8 3 1 3 10 9 3 8 1 3 5 5 4 9 10 1 3 5 10 6 1 1 6 5 10 8 10 8

1468014 1601469 1747057 1905880 2079141 2268153 2474348 2699288 2944677 3212375 3504409 3822991 4170535 4549674 4963280 5414487 5906713 6443687 7029476 7668519 8365657 9126171 9955823 10860897 11848251 12925364 14100397 15382251 16780637 18306149 19970344 21785829 23766359 25926937 28283931 30855197 33660215 36720234 40058437 43700113 47672850 52006745 56734631 61892324 67518899

46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90

R

n

91 0 93 0 95 0 97 0 99 0 101 0 103 0 105 0 107 0 109 0 111 0 113 0 115 0 117 0 119 0 121 0 123 0 125 0 127 0 129 0 131 0 133 0 135 0 137 0 139 0 141 0 143 0 145 0 147 0 149 0 151 0 153 0 155 0 157 0 159 0 161 0 163 0 165 0 167 8 169 5 171 6 173 0 175 0 177 0 179 0

810226780 883883759 964236823 1051894712 1147521496 1251841624 1365645402 1489794977 1625230882 1772979142 1934159062 2109991697 2301809125 2511064501 2739343090 2988374279 3260044667 3556412365 3879722572 4232424625 4617190501 5036935090 5494838272 5994369018 6539311647 7133794525 7782321294 8489805049 9261605507 10103569641 11022075966 12024082873 13117181308 14309652335 15610529814 17029668887 185 77820602 20266713379 167493499 182720180 199331105 217452114 237220489 258785989 282311986

k:

50

k:

R

n

92 94 96 98 100 102 104 106 lOB 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180

1 5 4 8 8 6 7 2 2 2 7 10 10 2 1 1 10 8 10 10 2 8 6 9 10 6 10 1 3 6 10 8 1 6 1 2 5 0 0 0 0 10 10 2 10

73656980 80353069 87657893 95626792 104320136 113803784 124149582 135435907 147748262 161179922 175832642 191817427 209255375 228278591 249031190 271670389 296367697 323310215 352702052 384765875 419744591 457903190 499530752 544942638 594482877 648526775 707483754 771800459 841964137 918506331 1002006906 1093098443 1192471028 1300877485 1419139074 1548151717 1688892782 1842428489 2009921980 2192642155 2391973254 19768374 21565499 23525999 25664726

Table 6. Some Wondrous Numbers and their Genetic Relations, 0=11

I 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

n

k

p

c

#

n

k

p

c

I

1345680 1468014 1601469 1747057 1905880 2038440 2079141 2223752 2268153 2425911 2474348 2597542 2646448 2699288 2833682 2887034 2944677 3091289 3149491 3212375 3372315 3435808 3504409 3606876 3678889 3748154 3822991 3934773 4013333 4088895 4170535 4292479 4378181 4460612 4549674 4682704 4776197 4866122 4963280 5108404 5210396

1127 1125 1123 1121 1119 1170 1117 1168 1115 1166 1113 1168 1164 1111 1166 1162 1109 1164 1160 1107 1162 1158 1105 1213 1160 1156 1103 1211 1158 1154 1101 1209 1156 1152 1099 1207 1154 1150 1097 1205 1152

8 8 8 8 8 7 8 7 8 7 8 8 7 8 8 7 8 8 7 8 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8

0 0

42

43

5308496 5414487 5572804 5684068 5791086 5906713 6079422 6200801 6317548 6443687 6632096 6764510 6891870 7029476 7235013 7379465 7518403 7668519 7892741 8050325 8050330 8201894 8365657 8610263 8782172 8782176 8782178 8947520 9126171 9393014 9580551 9580555 9580557 9760931 9955823 10057558 10246920 10246924 10451510 10451514 10451516

1148 1095 1203 1150 1146 1093 1201 1148 1144 1091 1199 1146 1142 1089 1197 1144 1140 1087 1195 1142 1142 1138 1085 1193 1140 1140 1140 1136 1083 1191 1138 1138 1138 1134 1081 1193 1189 1189 1136 1136 1136

7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 7 8 8 7 8 7 8 8 8 7 8 7 8 8 8 7 8 8 7 7 8 8 8

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 2 0 0 1 1 3 2 0 0 1 1 3 2 0 0 4 2 1 1 3 2

83 84 85

0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 1

0 0 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1

44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82

51

86 87 88 89

90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123

n

Ie

p

10648288 10860897 10971881 11178458 11178462 11311510 11401647 11401651 11401653 11616314 11700624 11735002 11735026 11848251 11969324 12194681 12194685 12339829 12438160 12438164 12438167 12438192 12672342 12764317 12801820 12801846 12925364 13057444 13303288 13303292 13461631 13568901 13568906 13568909 13568928 13568936 13824373 13924709 13965621 13965648 13965650

1132 1079 1191 1187 1187 1230 1134 1134 1134 1130 1077 1242 1242 1077 1189 1185 1185 1228 1132 1132 1132 1132 1128 1075 1240 1240 1075 1187 1183 1183 1226 1130 1130 1130 1130 1130 1126 1073 1238 1238 1238

7 8 8 7 7 3 8 8 8 7 8 7 7 8 8 7 7 3 8 8 8 8 7 8 7 7 8 8 7 7 3 8 8 8 8 8 7 8 7 7 7

C 0

0 4 2 1

0 1 3 2

0 5 3 4

0 4 2 1 0 1 3 2 6 0 5 3 4 0 4 2 1 0 1 3 2 7 6 0 5 3 5 4

Table 7. The 7-family. Grandparent = 2038440

#

1 2

N-child

3606876 1024692

N-childone step one before merging 293535231 381088411

N-parent one step before merging 2223752 2887034

K

K

c

p

Nmerger

N-max

K m

3

1173500

764423308

5791086

46 28 97

25

69493028

8601368512990

591

4

1173502 1396564

293535231

2223752

75

3

26685021

860 136851 2990

591

293535231

2223752

71

3

26685021

860 1368512990

587

N-max

K

5

3 9

26685021 34644401

8601368512990 8601368512990

538

562

Table 8. The 8-family. Grandparent = 1345680

#

N-child

N-childone one step before merging 504634735 326613848

No-

one before merging 3822991 2474348

2

259754 805033

3

878217

388697375

2944677

4

100575

504634735

3822991

5

117006

38236

290

6

124381

177629694

1345680

7

135689

211393897

1601469

K

K

c

p

Nmerger

m

25 15

45875885 29692168

22512799837 22512799837

489 463

32

19

35336125

91

45875885

22512799837 22512799837

514

1048

25 1098

3476

43996530071824

348

6

1

16148154

22512799837

8

5

19217627

22512799837

453 451

66 30

461

4. Iterating d(n) and o(n) - Two problems proposed by F. Smarandache

(a)

Let n be a positive integer and d(n) the number ofpositive divisors of n including 1 and n. Find the (smallest/ kfor which d(d(...d(n) .. .))=2 after k iterations, i.e. find k so that d(k)(nJ = 2.

den) is an important arithmetic function. We will look at its most important properties.

The factors of po., where p is a prime, are 1, p, p2, ... po.. Consequently d(pa.) = l+a.

The number of factors in n = p""p~ is easily seen to be (I +a XI +/3) from which the following important theorem follows:

I

In fact k is a (single-valued) function ofn.

52

If nl and n2 have no common divisor, i.e. are relatively prime which we write (nl,D2)=1, then d(nl,D2)=d(nl)d(n2). We say that the arithmetic function d(n) is multiplicative.

With n \\oritten in standard fonn n=PI '1>l ... p,' we have den) = (I+aXI+I3) .. 路(1+t)

(1)

We can now state: dCn)<n for all n. dC n)= I if and only if n= 1. dCn)=2 if and only ifn is a prime number.

From the above properties we see that dCn) is a measure of how far n is from being a prime number. The larger the number of factors of n the larger is dCn), dCn) being equal to 2 only when n is a prime. This makes it interesting to try to answer Smarandache's question [1]: How many iterations k are required in d(kj{n)=2?

Before looking at this problem let's make an important observation:

Given an arbitrarily large positive integer k we can always construct infinitely many integers n for which d(ij{n) > 2 for all i<k and d(k)(n) = 2.

Here is how. Let PI, Pz, '" Pk be odd primes (not necessarily distinct). Make the tollo\\oing series of constructions:

d(nJ) = 2 d(nJ) = PI

nI = PI

n2

= pPl-I 2

n3 =

P;:-

PI-Ill

pj2 -

n4

=

nk

= pnk-l-I

pn3-1

4

=

= pj'-I d(n4 ) = n3 d(n3)

k

So that for n = Ilk we have docJCn) = 2 while <i(iJCn) > 2 for i < k. Since we can choose our primes anyway we like as long as Pi 'f:. 2 this construction can be carried out in

53

infinitely many ways. If we ask for the smallest k for which d(kj{n) the answer is that such a k does not exist.

=

2 for all n then

Now to the problem. Factorizations and applications of (l) have been used to calculate k as a function of n for n:s:lOO, table 9. This does not tell us much about the general behaviour of doc)(n). Table 10 provides some interesting cumulative statistics for n :S 6 10 No more than 6 iterations are required for any n :S 10 6. It seems strange that the column for k = 7 remains empty for n :S 105 and n :S 106 in particular in view of the regular behaviour in the column for k = 3 and our previous observation that k can be arbitrarily large for properly chosen sufficiently large n. This calls for fiuther study.

Table 9. K as a function of n for n 5. 100 n

k

1 1 2 1 3 1 4 2 5 1 6 3 7 1 8 3 9 2 10 3

n

k

n

k

n

k

n

k

n

k

n

k

n

k

n

k

n

k

11 12 13 14 15 16 17 18 19 20

1 4 1 3 3 2 1 4 1 4

21

3 3 1 4 2 3 3 4 1 4

31 32

1 4 3 3 3 3 1 3 3 4

41 42 43

1 4 1 4 4 3 1 4 2 4

51 52 53 54 55 56 57 58 59 60

3 4 1 4 3 4 3 3 1 5

61 62

1 3 4 2 3 4 1 4 3 4

71

1 5 1 3 4 4 3 4 1 4

81 82

2 3 1 5 3 3 3 4 1 5

91 92 93 94 95 96 97 98 99 100

3 4 3 3 3 5 1 4 4 3

22 23 24 25 26 27 28 29 30

33 34 35 36 37

38 39

40

44 45 46 47 48 49 50

63 64 65

66 67

68 69 70

72 73 74 75 76

77 78 79

80

83 84

85 86 87

88 89 90

Let p# denote the product of all prime numbers less than or equal to p and consider the largest number r of distinct prime numbers which are needed to construct any integer :S lOS i.e. p,# :S lOS < p..l#. With these primes consider all possible constructions (2)

This does not mean constructing all n :S lOS but it does mean arriving at a structure into which all prime factorizations of n :S lOS fits. This will be so because any number :S lOS not produced by (2) will have fewer prime factors and smaller powers than one or more of the integers produced by (2). To illustrate this let's look at the case n :S 100. We have 2·3·5 < 100 < 2·3-5-7. We will therefore consider all possible construction 2o.·3~·5· :S 100. These are obtained for a :S 6, P :S 4 and 1: :S 2 resulting in table 11.

54

Table 10. Number of iteration k required to arrive at d(k)(n)=2 k=1

n~

10

k=2

4 25 168 1229 9592 78498

100 1000 1CXXXl 100000 1000000

k=3 2 7 16 33 79 189

3 34

348 3444 34429 344238

k=4

k=5

k=6

28 323 3181 30466 292460

5 144 2108 24839 271971

k=7

4 594 12643

Table 11. All possible prime factorization combinations C for n

~

100

#

C

d

n

t

C

d

n

t

C

d

n

t

C

d

n

1 2 3 4 5 6 7 8 9 10

000 001 002 010 011 012 020 021 030 040

1 2 3 2 4 6 3 6 4 5

1 5 25 3 15 75 9

11 12 13 14 15 16 17 18 19 20

100 101 102 1 10 111 120 121 130 200 201

2 4 6 4 8 6 12 8 3 6

2 10 50 6

410 500 510 600

10

48 32 96 64

30

9 6 12 9 4 8 8 12 5 10

31 32 33 34

20

202 210 21 1 220 300 301 310 320 400 401

100 12 60 36 8

18 90 54 4

21 22 23 24 25 26 27 28 29

45 27 81

30

6 12 7

40 24

72 16

80

Any number

:0; 100 corresponds to one or more of these structures, for example 77 = 7路11 corresponds to 1 1 0, 1 Oland 0 1 1. This means that d(n) can only assume values listed in table 11 for n :0; 100. The above scheme has been computer implemented for n:o; 10 12 . The result is sho\\TI on table 12 and figure 1, \\IDch gives a clear picture of the overall behaviour of d(n).

Finally we will be able to say something about drkj{n). For n :0; 10 12 we have d(n) :0; 6720<10 4 From table 10 it is seen that k :S 6 for n :S 104 and we therefore conclude that

55

Table 12. Largest value of d{n) for n!> 10k , k!> 12 k

Largest d

1 2 3 4 5 6 7 8 9 10 11 12

Number of

Number of

Prime comb.

Corresponding

d values combinations

for largest d

n

11 121 3111 311110 3311010 4211101 63111100 33211110 621111110 5312111100 6322111100 64211111100

6 90 840 9240 98280 942480 8648640 91891800 931170240 9777287520 97772875200 963761198400

4 12 32 64 128 240 448 768 1344 2304 4032 6720

4 11 22 38

4 34 141 522 1848 6179 20198 42950 133440 399341 783061 2309712

60 94 135 190 266 359 481 626

log (d)

1.3863 2.4849 3.4657 4.1589 4.8520 5.4806 6.1048 6.6438 7.2034 7.7424 8.3020 8.8128

10

8

6 Ir(d)

4

2

0

2

4

6 k

Diagram 1. Largest value of In(d(n)) for n < J(f.

56

8

10

12

(b) Let

L d and m a given positive integer. Find the smallest k for

CY(n) =

d'n.d>O

which a(a(... a(J) .. .))2m after k iterations, i.e. CY(k) (2) 2 m.

Clearly k is a fimction of m. It is a stepwise increasing fimction. The fIrst iterations have been used to illustrate this for the interval 2 ~ m ~ 25 in diagram2.

SIX

However, a far more interesting fimction to study is the inverse ofk(m). This fimction m(k) is growing so rapidly that numerical results are difficult to interpret and represent. A better picture of the behavior of cr1:(2) is obtained by studying the fimction f(k) = In(m)/k This fimction is represented in diagram 3 for the interval 1 ~ k ~ 100. After going through an interesting minimum for small values of k the curve flattens out. It seems to remain do\\'nwards convex. Does it approach a horizontal as)mptote'J Finally, a few iteration results (k,m): (1,3), (2,4), (3,7), (4,8), (5,15), (6,24), (7,60), (100,2972648508456959686477689735325484246606843303655482359755571200)

5

5

15

10

m Diagram 2. k as a function ofm

57

20

25

trk) 1

0.5

+---'-----+-----+------4------+------\ 20

40

60

100

k

Diagram 3. frkj=ln(m/k.

References:

J. F. Smarandache. Unsolved problem: 52. Only Problems..VOI Solutions, ISBN 1-879585-

00-6, Xiquan Publishing House, 1993

58

Chapter IV Diophantine Equations 1. Some Thoughts on the Equation lyP-x~=k

In his book Only Problems. Sot Solutions Smarandache formulated unsolved problem #20 as a conjecture rather than a problem: Let k be a non-zero integer. There are only a fInite munber of solutions in integers p, q, x, y, each greater than 1, to the equation x P - yq = k. I

In this study we will only consider the equation for p;tq. By writing the equation in the form IxP - yql = k we only need to consider cases where p > q.

For a given set of parameters (p,q,k:) it would then be desirable to list this fInite number of solutions (x,y). However, if this were possible if would probably already have been done. So Smarandache's statement is likely to be based on statistical evidence rather than analytical reasoning. It is mainly from the statistical point of "iew we will study this equation. The parameters will be restricted to ~()() and ~9. AB in most studies of this nature Ubasic provides the most effective computer language. All solutions where x<lOO and y<lOO can be churned out in a couple of seconds. To go any further we need a general approach to avoid running through meaningless search intervals. Consider

x=~yq+k For sufficiently large y only x =

r{ll or x = L{l J can produce solutions

corresponding to

1 Smarandacne

adds that "For k=1 this was conjeaured by Cassels (1953) and proved by Tijdeman 1976)." (Gamma 2/1986)

59

Chapter IV Diophantine Equations 1. Some Thoughts on the Equation lyP-x~=k

In his book Only Problems. Sot Solutions Smarandache formulated unsolved problem #20 as a conjecture rather than a problem: Let k be a non-zero integer. There are only a fInite munber of solutions in integers p, q, x, y, each greater than 1, to the equation x P - yq = k. I

In this study we will only consider the equation for p;tq. By writing the equation in the form IxP - yql = k we only need to consider cases where p > q.

For a given set of parameters (p,q,k:) it would then be desirable to list this fInite number of solutions (x,y). However, if this were possible if would probably already have been done. So Smarandache's statement is likely to be based on statistical evidence rather than analytical reasoning. It is mainly from the statistical point of "iew we will study this equation. The parameters will be restricted to ~()() and ~9. AB in most studies of this nature Ubasic provides the most effective computer language. All solutions where x<lOO and y<lOO can be churned out in a couple of seconds. To go any further we need a general approach to avoid running through meaningless search intervals. Consider

x=~yq+k For sufficiently large y only x =

r{ll or x = L{l J can produce solutions

corresponding to

1 Smarandacne

adds that "For k=1 this was conjeaured by Cassels (1953) and proved by Tijdeman 1976)." (Gamma 2/1986)

59

(la) or

(lb) It is easy to imagine from (I a) and (I b) that the number of solutions thin out rapidly as we increase y. Let's illustrate this for p=3 and q=2 by looking at the number of squares s which fit into the interval between (n_I)3 and (n+l?, i.e. integers s for which

(2) Let Smm be the smallest and Smax the largest s which satisfies (2). We can then calculate the ratio f between the number of squares between (n-l and (n+ I and the difference between these cubes, i.e.

i

Smax -

f = (n + 1)3 -

i

smin

(n _ 1)3

To have a solution we must have an s such that Is2 - n31 = k. The smaller f is the smaller is the chance for this to happen for our very limited range for k. Let S1 2be the largest square which is smaller than n 3 and sl be the smallest square which is larger than n3 then S1 and S2 give an indication of the behaviour of Is2 - n31 . These two ways of looking at the problem are displayed in table I for a sequence of values of n which have been chosen so that n is neither square nor cube.

Table 1. Frequency of squares

n

s2..n3

f13-s2

f

5

19 316 14761 430916 2515600 287598881 11203852544 513527672836 1151976475001

4 391 7600 276191 19845079 419507900 11156827231 193579108351 21208703299996

3.95·1(}2 1.40·1(}3 4.4Q. 1(}5 1.41·1(}6 4.47·1(}8 1.41·1(}9 4.47·1(}11 1.41·1(}12 4.47·1(}1'(

50 5·1()2 5· 1()3 5·10< 5· 1()5 5·1()6 5·10' 5·1 ()8

60

Table 2. Largest y

k 24 199 28 60 24 13 127 37 104 32 95 10 95 96 64 161 40 175 13 192 128

83 169 113

p

a

v

x

3 4 4 5 5 5 6 6 6 6 7 7 7 7 7 8 8 8 8 8 8 9 9 9

2 2 3 2 3 4 2 3 4 5 2 3 4 5 6 2 3 4 5 6 7 2 3 4

8158 10 15 76 4 3 4 2 3 2 6 3 6 2 2 3 2 2 2 2 2 3 2 2

736844 99 37 50354 10 4 63 3 5 2 529 13 23 2 2

80 6 3 3 2 2 140 7 5

Although the search for solutions was extended to y= 10000 no solutions were found for the following values of k: 6, 14,21,29,34,42,43, 50, 52, 58, 59, 62, 66, 69, 70, 75, 78, 82, 85, 86, 91, 102, 110,111,114,123,125,130,133,134,146,149,150,158,160,165, 173, 176, 177, 178, 182, 187, 189, 195.

No solutions were found for (p,q)=(9,5), (9,6), (9,7) and (9,8).

Computer search: Calculations were carried out in Ubasic for y~I0000, lcQOO and ~9. Most solutions occur for small values ofy for which we cannot use the CEIL l) and the FLOOR cL J) functions. The ROUND function has been used instead.

Calculations are carried out with real numbers to 19 decimal places (POINT 9 in the Ubasic language).To safegard against "near integer" solutions a "proposed" solution is

cr

61

recalculated with integers in a subroutine. A simple version of the program is given beIO\,,~

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240

point 9 Y1=10000 for R%=3 to 9 for S%=2 to R%-l for K%= 1 to 200 Y=2 while Y<Y 1 if (YI\R%-K%)<l then X1=(K'7'o-YI\R%)1\(1/S%):goto 100 X1=(YI\R%-K%)1\(1/S%) X=round(X1) if abs(X-X1)<101\(-30) then gosub 210 endif X1=(YI\R%+K%)1\(1/S%) X=round(X1) if abs(X-X1 )<101\(-30) then gosub 210 endif inc Y wend next:next:next end if abs(YI\R%-XI\S%)<>K% then goto 240 if X<=l then goto 240 print R%,S%,YXK% return

The number of solutions for each set of parameters is given in table 3. The largest value of y which occurs in a solution for each parameter set (p,q) is given in table 2, which confmns the indications for the rarity of solutions for large y given in table 1.

The largest number of solutions (11) occur for k= 17. Several of these are due to the fact that no distinction is made between (X路)b and (xb)". These solutions are displayed in table 4. A limited search (y:o;lOO) was carried out for lO:O;p::;20, kQOO. Only two solutions with y;102 were found. These results are shown in table 5.

Conclusion: Smarandache's conjecture is well supported by the numerical results obtained in this study. The number of solutions diminish rapidly with increasing y, p andq.

62

Table 4. Solutions for k=17 Sol. #

P

q

Y

1 2 3 4 5 6 7

3 3 3 3 3 4 4 5 6 6 9

2 2 2 2 2 2 3 2 2 4 2

2 4

8 9 10 11

8 43 52 3 3 2 2 2 2

)II

5 9 23 282 375 8 4 7 9 3 23

Table 5. Solutions for 1~p$20

Ie:

p

q

y

63 65 124 132 183 24 23

10 10 10

2 2 2 2 2 3 2 2 2 2 2 2 3 7 2 2 2 2 3 2 2

2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3

68 94 112 161 199 149 139 127 129

89 92 192 7 37

10 10

10 11 11 11 11

11 11 11 11 12 12 13 13 13 15 15

63

)II

31

33 30 34 29 10 45 46 421 44 47

43 13 3

63 65 91

90 20 181 3788

Table 3a. The number of solutions to IYI'-xql=k

p

3

4

4

5

5

5

6

6

6

6

7

7

7

7

7

8

8

8

8

8

8

9

k/q

2

2

3

2

3

4

2

3

4

5

2

3

4

5

6

2

3

4

5

6

7

2 Sum

1 2 3 4 5 7 8 9

1 1 1 3 1 5 3 4 1 4 2 4 2 3

2 1 2 3

2

10 11 12 13 15 16 17 18 19

20 22 23 24 25 26 27 28

30

2

5 2 2 1 1 1 2

36 37

38 39 40 41 44

3 4 2 2 2 4 1 2 1 9 1 2 5 3 2 3 3 1 4 5 2 3

2

4 1 1 1 2

31 32

33 35

11

2

1

1 2 1 1 3 2 1 2

64

Table 3b. The number of solutions to lyp-xQI=k

p

4

4

5

5

5

6

6

6

6

7

7

7

7

7

8

8

8

8

8

8

9

k/q 2 2

3

2 3

4

2 3

4

5

2 3

4

5

6

2 3

4

5

6

7

2 Sum

3

2

45 46 47

3

48

2

49 51 53 54 55 56 57 60 61 63

1

1 6 6 4 1 3 3 5 4 3 5

2 2 2

3

2 2 1 1

2 2 5 4 1 4 5

64

2

65 67

68 71 72 73 74 76

2

2

1 4 1 1

6

2

2 2

1 1 3 3

1 1

2 2

77 79

80 81

83 84

1

87

88 89 92

1

1 5

2 4

3 1

2 2

93 94 95 96 97 98

1 3

2 3 1 ')

99

65

Table 3co The number of solutions to Iyp-xql=k p

3 4 4 5 5 5 6 6 6 6

7 7 7 7 7 8 8 8 8 8 8 9 9 9

k/q 2 2 3 2 3 4 2 3 4 5 2 3 4 5 6 2 3 4 5 6 7 2 3 4 Sum 100 101 103 104 105 106 107 108 109 112 113 115 116 117 118 119 120 121 122 124 126 127 128 129 131 132 135 136 0/37 138 139 140 141 142 143

8 2 1 7 5 2 1 1 2 9 7 2 3 2 3 2 1 2 1 3 1 3 6 3 3 3 6 1 1 1 1 1 2 1 2 4 3

6 1 3 1 1 1 2 2 1 1 1 2 4 1 3 1 1 1

1 1

2

2 1 1 2 1 1 1 1 1 3 1

144

2

145 2

66

Table 3d. The number of solutions to Iyp-xql=k

p

3

4

4

5

5

5 6 6 6 6 7 7 7 7 7 8 8 8 8 8 8 9 9 9

k/q 2 2 3 2 3 4 2 3 4 5 2 3 4 5 6 2 3 4 5 6 7 2 3 4 Sum 147 148 151 152 153 154 155 156 157 159 161 162 163 164 166 167 168 169 170 171

i72

1

1

,

6

2

3 3

1 2

4 1 1 1

1 2 1 1 2

174 175 179 ~30

2 2 2 4 3 1 1 3 1 2

2 1 1 3 1 1

4 1 2 2 1 5 3 3 1

2 2

i ~3 34 ,85 '86

4

2

188 2 190 191 192 2 193 194 196 197 1 198 2 199 2 200 3

2 1 3 1 1 9 1 1 2

1

4 3

6

67

This equation fIrst appeared in the Numbers Count column of the Personal Computer World in March 1986. At that time the author found two until then unkno",n solutions (p,q,r,s,t)=(87,42,6,3,3) and (p,q,r,s,t)=(99,97,39,13,2). This took approximately 75 hrs on an 8086 processor running a 4.7 Mhz. Now, in 1996, these results were reproduced using the same program in 21 minutes on a Pentium 100 Mhz. The method used is reproduced below together with a number of new solutions.

Consider the function

This function is invariant under exchange of variables. This makes it useful to study the function ",rule all variables except one is kept constant. Denote this variable x.

After some elementary algebraic manipulations we have F( ... ,x+l, ... ) = F( ... ,x, ... )+ G(x) -lOCH(x)+ D/x where G(x) = 28x3 + 22x2 + 8x + 2 H(x) = 2x + 1 C = p2+q2+r+s2+t2 with one of the constants replaced by x D = 90pqrst with the same constant as above replaced by x This permits us to calculate F( " . ,x+ l, ... ), G( x), H( x), C and D. When, in the next step, one of the unknowns p,q,r,s,t is increased by one the following replacements must be made C:= C + H(x) D := D(x+ l)lx

Without imposing any restrictions we can assume p ;?: q ;?: r;?: s ;?: t. For a given value of p the search will be conducted for descending values of the other unknowns. For given values of p, q, f, S consider the function

68

We have g'(t) = 8f - 20at +b g"(t) = 24t2 - 20a g"(t) = 0 for tm = ..J(5a/6) which gives g'min = b - 161m3 Case 1: g'min > o. Since g'(O»O it follows that g'(t»O for t>O. This means that get) is an increasing function for t>O If we have found tl such that g(tl»O then g(t»O for all t>tl and the search can be interrupted for t=tl.

Case 2: g'~O. For t>1m the function get) is convex and a value t=tl will be found for which the function is positive and increasing. The search is stopped for t>tl>1m.

The above method has been used in a computer program written in Ubasic.

Implementation on Pentium 100 Mhz computer has revealed a few new solutions. A complete list of all solutions for p9tOO is given in table 4.

Table 4. Solutions for p:S:400

#

P

9

1

1

1

2

2

3

2

2

1

4

3

3

2

5

4

2

6

6

3

2

1

7

7

7

4

2

8

17

7

7

1

1

9

59

47

19

7

2

10

87

42

6

3

3

11

99

97

39

13

2

12

124

63

42

17

13

127

47

34

2

1

14

189

87

27

3

3

15

286

154

11

11

11

69

3.

The Equation y =

2'XtX2 ••• XI<

+1

Conjecture:

Let le2 be a positive integer. The diophantine equation: y = 2'XIX2 ... Xk + I has a infInitely many solutions in distinct primes y, Xl,X2, ... Xk. This is unsolved problem number 11 in Smarandache's book Only Problems, Not Solutions. The word distinct has been added by the author. The purpose of this study is to see 'how stable' this conjecture is. This is done through a computer analysis of all possible solutions for y:5 109 (A very thin layer when surfmg the ocean of integers but big enough to take a bit of time on the computer when it comes to calculations -in fact more than 100 hrs). From the computational point of view there is no reason to exclude k= I. The interval O<y<10 9 is divided into 10 sub-intervals of equal length; 0<v<108 Interval # 1: 1Q8<v<2·1Q8 Interval #2: Interval #3: 2.IQ8<y<3.IQ8 Interval # 10 The endpoints are excluded since these do not contribute to the number of solutions.

Consider t=(y-I )/2='XIX2 ... Xk

(I)

The task is to identifY sequentially all square free numbers n<109 For each square free number with k distinct prime factors we calculate the corresponding number y and test whether it is prime or not. The number of primes is denoted II1k and the number of square free numbers is denoted Ilk. mk and Ilk are recorded for each interval and the frequency of solutions Fk=rIlJc/IIk is calculated. The result is shown in table 5.

The same result are shown in diagram I, which conveys a good visualization of a result obtained through surfmg on a small area of the ocean of integers.

Let's make a few observations: Why the irregularities for k=8 and k=9? The smallest square free integer "'ith k prime factors is Pk# where Pk is the kth prime. For k=8 and k=9 this means there can be no solutions for y<2·19#+1=19399381 and y<2·23#+1=446185741 respectively. The samples for k=8 and k=9 are therefore too small to give a true picture - randomness takes precedence over mass behaviour.

70

Table 5. Frequency of solutions

#

k=!

k=2

k=3

k=4

k=5

}(=6

k=7

k=8

I

0.2435

0.3419

0.2167

0.2607

k=9

0.0765

0.0893

0.1061

0.1282

0.1561

2

0.0701

0.0817

0.1159

0.1407

0.1933 0.1735

3

0.0683

0.0794

0.1118

0.1356

0.1665

0.2075

0.2810

4

0.0672

0.1095

0.1323

0.1612

0.0662

0.1301

0.1580

6

0.0655

0.0760

0.1080 0.1063

0.1991 0.1944

0.2622

5

0.0778 0.0769

0.1280

0.2298

1.0000

0.0653

0.0752

0.1054

0.1266

0.1562 0.1543

0.1944

7 8 9

0.1898

0.2496

1.0000

0.0646

0.0748

0.r:l;67 0.0937 0.0919 0.0903 0.0895 0.0887 0.0878

0.1046

0.1256

0.1529

0.1846

0.2225 0.5000

0.0641

0.0744

0.0873

0.1036

0.1244

0.1508

0.1883

0.2359 0.6667

10

0.0639

0.0738

0.0867

0.1030

0.1238

0.1498

0.1846

0.2120 004000

Frequency of solutions

1.0 0.8 0.6

F 0.4

k=9

0.2

k

0.0 C')

l!)

I'-

k=1 m

Interv 01

Diagram 1 Frequency of solutions

71

0.2252 0.0000

With the previous observation in mind let's compare the frequencies for the fIrst and the last interval. Diagram 2 shows Fk for the intervals 0<y<108 and 9·108<y<109 .

Frequency of solutions for intervals 1 and 10

0.4 !

0.35

•

0.3 0.25 i u..

I 0.15 0.2

I

0.1 O.as

1

i

I

~

,

!

• • ••

• ..

-10 i

0 1 0

2

4

6

.1

8

•

10

k

Diagram 2

Observation: When not too close to pJI the frequency smoothly increases with increasing k. TIlls is a good support for Samrandache's conjecture.

Let's now take a closer look at how the frequency of solutions behave for distinct values of k. Diagram 3 shows that the frequency decreases slightly for all k as larger integers as included. Is there an asymptote for each k? The frequency of solutions increases as we increase k. It should be noted that it is the ratio between the number of solutions and the number of square free numbers in an interval determined by t in (1) \\<IDch depicted. TIlls is of course different from the number of square free numbers in the interval for y because when y runs through the interval a<y<b then we consider square free numbers s which obey (a-I )/2<s«b-I )/2

72

As a bi-product to this study we have information on how many square free integers

with 1, 2, .... , 9 prime factors there are in our different intervals. This is shO\\TI in table 6, where, as we have seen, we now only have five intervals:.

O<s<108 Interval #1: Interval #2: 108<s<2·108 Interval #3: 2·10 8<s<3·10 8 Interval #4: 3· 108<s<4· 108 Interval #5: 4. 108<s<5· 108

Frequency of solutions by intervals 0.4

--+-k=1 -k=2

0.3

~k=3

----*-k=4

F 0.2

-.-k=5 --+-k=6

0.1

~k=7

-k=8 0.0

2

0

4

6

8

10

Interval #

Diagram 3.

Table 6. Number of square free numbers #

1<:=1

1<:=2

1<:=3

1<:=5

1<:=4

5761455 17426029 20710806 12364826

3884936

k=6

1<:=7

605939

38186

1<:=8

1<:=9

516

0

2

5317482 16565025 20539998 13029165

4476048

799963

63642

1409

0

3

5173395 16270874 20457818 13243252

4689541

879765

76114

2060

2

4

5084001

16085983 20402004 13374830

4825914

932513

84968

2560

6

5

5019541

15951738 20359052 13468885

4926227

972398

91767

3005

8

73

Diagram 4 shows a cumulative representation of the number of square free numbers with k=l. 2, 3, ... 9 prime tactors. A square free number \\ith only one prime factor is indeed a prime number so the fIrst column in table 6 shows the number of prime numbers in the interval.

Cumulative statistics of square free numbers

1~r-------------------------------------~

05

.4 03 112 01

k= 1

k=2

k=3

k=4

k=5

k=6

k= 7

k=8

k=9

Diagram 4.

The illustration on the cover is another version of the above diagram for n<5路IQc. Finally, a problem often leads to other problems. Let's go back to iterations. In the introduction to chapter III it was said that iterations result in an invariant a loop or divergence. Is there really no other possibility? Let's look at this: We define Uk-! = 2路Uk+ I where UI is a prime number. Uk-: \\ill have the property of being a prime number or not being a prime number.

In this case we can give an explicit formula for ilk-I

= 2k(UI+I)-1.

74

ilk~l

in terms OfUI. One easily fmds

How do we characterize the result of iterations in relation to the property in which we are interested? Indefinite') How many times can we iterate Uk-I'" 2路Uk+ I preserving primality? For uI"'305192579 it is eight times resulting in the following series of nine primes:

305192579,610385159,1220770319,2441540639,4883081279, 9766162559,19532325119,39064650239,78129300479 Which is the first series with 10 terms - or 11') Maybe we need some deep sea diving~

75

The Smarandache Ceil Function Definition: For a positive integer n the Smarandache ceil fimction of order k is deified through t s.d:n) = m where m is the 5

"t

positive integer for which n divides mt..

In the study of this function we will make frequent use of the ceil fimction defmed as follows:

rxl = the smallest integer not less than x.

The following properties follow directly from the above defmitions:

1.

2.

3.

St(n) = n St{pa) = ~aIk1 for any prime number p.

For distinct primes p, q, ... r we have St<paqll...I) = ~alkl<lJllkl ... ./Ml.

Henry Ibstedt

$ 9.95