An experimental evidence on the validity of third Smarandache conjecture on primes Felice Russo
_\/tcr()J7 7ecil17()/o!!:1' 110/\' '- .-1 re~~CIl7() (-1(1) IW(1'
Abstract
In this note v.e report the results regarding ,he check of the third Smarandache conjecture on primes [1 J.[2] for P ~ 2 25 and 2 ~ k ~ 10, In the range analysed the conjecture is true, n
\loreover. according to experimental data obtained. it seems likely that the conjecture is true for all primes and for all positi\'e values ofk,
Jntrod uction
In [1] and [2] the follo\ving function has been defined:
\vhere Pn is the n-th prime and k is a positive integer \loreover in the above mentioned papers the follov. .-ing conjecture has been fomlUlated by F, Smarandache
2 C(n,k)< k for k~2 This conjecture is the generalization of the Andrica conjecture (k=2) [3] that has not yet been proven This third Smarandache conjecture has been tested for P ~ 2 25 . 2 ~ k ~ 10 and in this n
note the result of this search is reported The computer code has been written utilizing the Loasic software package
Experimental Results
In the follov,ing graph the Smarandache function for k=-1- and n< I 000 is reported As we can see the value ofC(k.n) is modulated by the prime-s gap indicated by d n = Pn+l - Pn We call this graph the Smarandache --comet"
38
Smarandache function for k=4, 1<=n<=1000 Smarandache comet
-=-1--0=-2-'1 0
,
~,
.
-.~..
C(
~
n,4 001 ) .
_
~
~
............. ~~:.:.
-d=6\1
I - 0=8
0=10!1 I' 0=14,1
d 12
I
.
•
'." --
-
-
eX
_____
-----.-
-
........................ --..-..:::::--- - - - ~:::---__ .,.~~~
-
-
'. ~------=---=::-IIS.~ ~ t
'1"1
'"
-
'1
'I
d=16
0=18
I
0=20
0=2211
I
-d=24
.0=26:1
1
d=28
d=30lil,
0=32
'1'1'
1
-__
.................
0.001 + - - - - - - - - - - - - - -
-
i: Ii
=
,
-
i
0 =4
I,
0.1~---------------------------
I;'
I
1r1.
I'
~~ ....... ~!
I ''1
1
X
II
ii "
, - I_ _ _
i!
~
0.~1+_--r_-~--_r--~-~--~--_r--~-~---_c
o
100
200
400
300
500
700
600
BOO
1000
900
n
In the following table, instead. we report •
the largest value ~\'lax_C( nJ..:) of Smarandache function for 2::::;; k : : ; 10 and p
•
the ditference .1(k) between 2/k and \lax_ C(n.k)
•
the value of Pn that maximize C( n.k)
•
the value of 11k
: : ; 2 25
n
k
2
3
4
5
6
7
8
9
10
;\'1ax C(n.k)
0.67087
0.31105
0.19458
0.13962
0.10821
0.08857
0.07564
0.06598
0.05850
~
0.32913
0.35562
0.30542
0.26038
0.22512
0.19715
0.17436
0.15624
0.14150
7
7
7
7
7
3
3
3
3
1
0.666 ..
0.5
0.4
0.333 ..
0.2857 ..
0.25
0.222 ..
0.20
Pn 2/k
According to previous data the third Smarandache conjecture is veritied in the range of k and Pn analysed due to the fact that ~ is always positive. \Ioreover since the Smarandache C(n.k) function falls asymptotically as that the estimated maximum is valid also for p
n
>2
39
25
.
f1
increases it
IS
likelv
We can also analyse the behaviour of difference I:!(k) versus the k parameter that following graph is showed with white dots We have estimated an interpolating function:
In
the
1 I:!(k) ~ 0.88· k . O 78 \\ith a \ery good R 2 \'alue (see the continuous CUf\e) This result reinforces the validit\· of the third Smarandache conjecture since
I:!(k) ~
°
II:! (k ) 0.40000 0.35000
for k ~ 00
vs
kl
y
=0.8829x..,·71102 R2 =0.9904
1 -'
c
0.30000
C
0.25000
It; (k ) I
0
0.20000
-~
I
c:
I
0.15000
i
"
I 0.10000
I
0.05000
I
0.00000 \ 3
---:J
I I
4
5
6
7 k
8
9
10
i I
I
I
~ew
Question
According to previous experimental data can we reformulate the third Smarandache conjecture v,ith a tighter limit as showed below'')
2 C(n,k)<-2·a k 0
where k
~
2 and a o is the Smarandache constant [-+ ].[ I ]
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References: [1] See http//v.vvwgallup.unm.edu/-smarandacheIConjPrim.txt [2] Smarandache, Floremin, "Conjectures which Generalize Andrica's Conjecture", Octogon, B ra.;;o v, Vol. 7, \:0 L 19C)C). 173-176
[3] Weisstein, Eric W." "Andrica"s conjecture"" in CRC Concise Encyclopedia of 1'v1athematics, CRC Press, Florida, 1998. [-+] Sloane, Neil. Sequence A038458 ("Smarandache Constant" 0.5671481302020177146468468755 ... ) m <An On-Line Version of the Encyclopedia of Integer Sequences>, http://ww\.v.research.art.comcgi-bin/access.cgi/asinjas/sequences/eishis.cgi.
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