A brief account on Smarandache 2-2 subtractive relationships

A brief account on Smarandache 2-2 subtractive relationships Henry Ibstedt

Abstract: An analysis of the number of relations of the type S(n} S(n+l)=S(n+2}S(n+3) for n<108 where S(n) is the Smarandache function leads to the plausible conclusion that there are infinitely many of those.

This briefnote on Smarandache 2-2 subtractive relationships should be seen in relation to the article on Smarandache k-k additive relationships in this issue of SNJ [1]. A Smarandache 2-2 subtractive relationship is defined by 8(n)-8(n+ 1)=S(n+2)-S(n+3) where Sen) denotes the Smarandache function. In an article by Bencze [2] three 2-2 subtractive relationships are given 8(1)-8(2)=S(3)-S(4), 1-2=3-4 8(2)-S(3)=8(4)-8(5), 2-3=4-5 S(49)-S(50)=S(51 )-S( 52), 14-10=17-13 The first of these solutions must be rejected since S(I)=O not 1. The question raised in the article is ''How many quadruplets verify a Smarandache 2-2 subtractive relationship?'

As in the case of Smarandache 2-2 additive relationships a search was carried for ~108. In all 442 solutions were found. The first 50 of these are shown in table 1.

Table 1. The 50 first 2-2 subtractive relations. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14

15 16 17 18 19 20 21 22

n 2 40 49 107 2315 3913 4157 4170 11344 11604 11968 13244 15048 19180 19692 26219 29352 29415 43015 44358 59498 140943

S(n) 2 5 14 107 463 43 4157 139 709 967 17 43 19 137 547 167 1223 53 1229 7393 419 4271

S(n+1) 3 41 10 9 193 103 11 97 2269 211 11969 883 149 19181 419 23 197 3677 283 6337 601 383

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S(n+3) 4 7 17 109 331 29 4159 149 61 829 19 179 43 139 229 2017 1129 1279 1103 1109 17 4027

S(n+4) 5 43 13 11 61 89 13 107 1621 73 11971 1019 173 19183 101 1873 103 4903 157 53 199 139

Table 1. continued.

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23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

n 147599 153386 169533 181577 186056 201965 204189 210219 217591 246974 253672 257543 262905 273815 321010 363653 407836 431575 451230 530452 549542 573073 589985 590569 608333 646333 649702 666647

Sin) 1433 283 23 971 1789 1303 2347 887 151 59 857 1801 1031 2381 683 163 31 283 89 202 2309 2909 631 353 1907 15031 577 666647

S(n+1)

S(n+3)

SinH)

41 23 79 571 2297 821 2917 457 461 89 167 73 211 3803 821 227 661 739 127 166 2207 2837 449 809 1913 24859 1447 197

2203 1237 827 1697 2269 1453 139 659 8059 227 829 2711 929 3299 241 283 673 607 239 419 941 283 1291 317 191 271 107 666649

811 977 883 1297 2777 971 709 229 8369 257 139 983 109 4721 379 347 1303 1063 277 383 839 211 1109 773 197 10099 977 199

As in the case of 2-2 additive relations there is a great number of solutions formed by pairs of prime twins.

Table 2. AIISI subtractive relations formed by pairs of prime twins for n<108. # 1 2 3 4 5 6 7 8 9 10 11 12 13 14

n 40 4157 11968 19180 666647 895157 1695789 1995526 2007880 2272547 3198730 3483088 3546268 4194917

S (n) 5 4157 17 137 666647 895157 347 71 101 2272547 1787 227 431 4194917

S(n+l) 41 11 11969 19181 197 137 101 1995527 2007881 149 3198731 3483089 3546269 197

100

S(n+3) 7 4159 19 139 666649 895159 349 73 103 2272549 1789 229 433 4194919

S(n+4) 43 13 11971 19183 199 139 103 1995529 2007883 151 3198733 3483091 3546271 199

Table 2. Continued t 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 42 43 44 45 46 47 48 49 50 51

n 4503640 5152420 6634078 6729658 7455628 7831738 7924877 11001647 11053978 12466690 13530988 17293120 17424707 18173650 19222600 19227910 22208567 26037491 30468670 31815238 36683147 40881257 42782236 46238236 53009681 53679671 53906597 54747418 57935326 63694847 68203229 73763380 84344411 86250580 92596529 94788077 95489237

sen) 179 149 269 107 2729 641 7924877 11001647 281 809 311 4157 17424707 191 431 419 22208567 26037491 311 5639 36683147 40881257 227 311 53009681 53679671 53906597 269 659 63694847 68203229 2381 84344411 1667 92596529 94788077 95489237

S(n+1)

S(n+3)

S{n+4)

4503641 5152421 6634079 6729659 7455629 7831739 71 239 11053979

181 151 271 109 2731 643 7924879 11001649 283 811 313 4159 17424709 193 433 421 22208569 26037493 313 5641 36683149 40881259 229 313 53009683 53679673 53906599 271 661 63694849 68203231 2383 84344413 1669 92596531 94788079 95489239

4503643 5152423 6634081 6729661 7455631 7831741 73 241 11053981 12466693 13530991 17293123 313 18173653 19222603 19227913 433 349 30468673 31815241 643 193 42782239 46238239 1063 523 229 54747421 57935329 1483 1723 73763383 271 86250583 1021 1033 103

124666~1

13530989 17293121 311 18173651 19222601 19227911 431 347 30468671 31815239 641 191 42782237 46238237 1061 521 227 54747419 57935327 1481 1721 73763381 269 86250581 1019 1031 101

In the case of2-2 additive relations only 2 solutions contained composite numbers and these were the first two. This was explained in terms of the distribution Smarandache functions values. For the same reason 2-2 subtractive relations containing composite numbers are also scarce. but there are 6 of them for n<108. These are shown in table 3.

It is interesting to note that solutions #3, #5 and #6 have in common with the solutions formed by pairs of prime twins that they are formed by pairs of numbers whose difference is 2. Finally table 4 shows a tabular comparison between the solutions to the 2-2 additive and 2-2 subtractive solutions for n<108. The great similarity between these results leads the conclusion: If the conjecture that there are infinitely many 2-2 additive relations is valid then we also have the following conjecture:

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Table 3. All 2-2 subtractive relations <108 containing composite numbers. t

n

5(n)

5(n+1)

5(n+3)

S(n+4)

1

2

2

3

4

5

2

49

14

10

17

13

3

107

107

9

109

11

4

530452

202

166

419

383

5

41839378

111

41839379

113

41839381

6

48506848

57

48506849

59

48506851

Table 4. Comparison between 2-2 additive and 2-2 subtractive relations.

Total number of solutions Number formed by pairs of prime twins Number containing composite numbers

Number of2-2 additive solutions 481 65 2

Number of2-2 subtractive sol 442 51 6

Conjecture: There are infinitely many Smarandache 2-2 subtractive relationships.

References H. Ibstedt. Smarandache k-k additive relationships, Smarandache Notions Journal, Vol. 12 (this issue).

2. M Bencze, Smarandache Relationships and Subsequences, Smarandache Notions Journal, Vol. 11, No 1-2-3, pgs 79-85.

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