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Primes in the Smarandache deconstructive sequence.

Abstract In this article, we present the results of investigation of first 100100 terms of Smarandache Deconstructive Sequence and report soiree new primes and other results found from the sequence.

Keywords The Smarandache deconstructive sequence. initial digits, trailing digits.

Introduction

The Smarandache Deconstructive sequence of integers (1] is constructed by sequentially repeating the digits 1 to 9 as follows:

1, 23, 456, 7891, 23456, 789123.......

Kashihara [2] asked: How many primes are there in the sequence. Ashbacher [3] explored this question and raised some more questions, which "were studied and answered by Henry Ibstedt [4].

Let its call the sequence mentioned above as Smarandache Deconstructive sequence of the first kind (SDS-1) because a similar Deconstructive sequence can be constructed by sequentially repeating the digits 0 to 9 as follows [4].

0, 12, 345, 6789, 01234, 561890,.....

The Smarandache Deconstructive sequence of integers [1] is constructed by sequentially repeating the digits 1 to 9 as follows:

1, 23, 456, 7891, 23456, 789123,......

Kashihara [2] asked: How many primes are there in the sequence. Ashbacher [3] explored this question and raised some more questions, which Naere studied and answered by Henry Ibstedt [4].

Let its call the sequence mentioned above as Smarandache Deconstructive sequence of the first kind (SDS-1) because a similar Deconstructive sequence can be constructed by sequentially repeating the digits 0 to 9 as follows [4].

0, 12, 345, 6789, 01234, 567890,.......

This can be termed as the Smarandache deconstructive sequence of the second kind (SDS-II).

In this paper, we report the primes found in both the sequence after checking the first 10000 terms of both these sequence.

This can be termed as the Smarandache deconstructive sequence of the second kind (SDS-II).

In this paper. we report the primes found in both the sequence after checking the first 10000 terms of both these sequence.

Primes in the Smarandache Deconstructive Sequence of first kind:

The following 13 primes in the Smarandache Deconstructive sequence of first kind have been reported earlier [5] [6].

23, 4567891, 23456789,......

These are 2, 7, 8, 10, 17, 20, 25, 28, 31, 38, 61, 62 and 355-th term of the sequence and are given in Table-1 below:

Note that [(123456789).sub.39] cleans 123456789 repeated 39 times. On further computation up to 10000 terms of the sequence, we have noted following 3 more primes, namely the term 4690, 4772 and 8162 of the sequence. Since the primality of the term 4690, 4772 and 8162 have not been certified, so these call be treated as probable primes. These are: [(123456789).sub.521]1, 23456789[(123456789).sub.529]123, 23456789[(123456789).sub.906].

It may be noted that though there are 12 primes in the first 62 terms of the sequence but only 16 primes in the first 10000 terms of the sequence. So the percentage of primes is reducing significantly which is in accordance with prime number theorem, according to which, the probability that a random chosen number of size n is prime decreases as 1/d (where d is the number of digits of n).

Observations on the Smarandache Deconstructive Sequence of first kind:

From the term of this sequence, it is seen that the trailing digit (units digit) repeats the pattern.

1, 3, 6, 1, 6, 3, 1, 9, 9; .....

Interestingly this sequence is the same as the sequence of digital root of triangular numbers.

Similarly initial digit of the element of SDS-I repeats the pattern 1, 2, 4, 7, 2 7, 4, 2, 1; .....

Table-2 below gives the possible combination of initial and trailing digits of any element of SDS-I.

Since the trailing digits of the term of the SDS-I sequence can be 1, 3, 6 or 9, it is obvious that for an element to be prime, the only possible trailing digits are 1, 3 or 9. If trailing digit is 3, possible initial digits are 2 and 7, but if initial digit is 7 and trailing digit is 3, the number is divisible by 3. Similarly if trailing digit is 9 and initial digit is 1, the number is divisible by 3. The possible combinations of trailing and initial digits for a prime in the sequence are given in Table-3.

So there are 5 possibilities out of 9, as the pattern repeat for every 9 elements in the sequence. Out of 13 primes found, 7 ends in 1, 3 ends in 3, 3 end in 9 and primes corresponding to all 5 possibilities are found. The three probable primes found end in 1, 3 and 9 respectively.

It is thus clear that the term 3 + 9n, 5 + 9n, 6 + 9n, 9n of the sequence are obviously composite and need not be checked for primality. Only the term 9n + 1, 9n + 2, 9n + 4, 9n + 7 and 9n + 8 need to be checked for primality.

Conjecture 1. Every prime except 5 divides some element of the sequence.

It is noted that none of the element of the sequence end in 0 or 5. So 5 cannot be a factor of any terms of the sequence. It has been checked that every prime up to 3821 except 5 divides some element of the sequence up to 10000 terms, so it is quite reasonable to conjecture that: every prime except 5 divides some element of the sequence. Can this be proved?

Primes in the Smarandache Deconstructive Sequence of second kind:

0n computation up to 10000 terms of the sequence, we have noted only 2 primes, namely the term 367 and 567 of the sequence. These are:

[(1234567890).sub.36]1234567, [((1234567890).sub.56]1234567.

Observations on the Smarandache Deconstructive Sequence of second kind:

From the terms of this sequence, it is seen that the trailing digit (units digit) repeats the pattern.

0, 2, 5, 9, 4, 0, 7, 5, 4, 4, 5, 7, 0, 4, 9, 5, 2 0, 9, 9;......

Similarly initial digit of the element of SDS-II repeats the pattern

0, 1, 3, 6, 0, 5, 1, 8, 6, 5, 5, 6, 8, 1, 5, 0, 6, 3, 1, 0;......

Table-4 below gives the possible combination of initial and trailing digits of any element of SDS-II

Since the trailing digits of the terms of the SDS-II sequence can be 0, 2, 4, 5, 7 or 9, it is obvious that for an element to be prime, the, only possible, trailing digits are 7 or 9. If trailing digit is 7, possible initial digits are 1 or 6. Similarly if trailing digit is 9, possible initial digits are 0, 1, 5 or 6. If initial digit is 0, 1 or 6 and trailing digit is 9, the number is divisible by 3. So it cannot be prime. The possible combinations of trailing and initial digits for a prime in the sequence are given in Table-5.

So there are 3 possibilities out of 20, as the pattern repeat for every 20 elements in the sequence. Both the primes in first 10000 terms of the sequence end in 7 and initial digit in both primes is 1. It remains to find a single prime corresponding to trailing digit 9 and also corresponding to trailing digit 7 with initial digit 6. The only terms needs to be checked for primality are 7 + 20n, 12 + 20n and 15 + 20n. It is interesting to note that for every 20 terms of the sequence, only 3 needs to be checked for possible primes, whereas in SDS-I, for every 9 terms of the sequence, 5 terms needs to be checked for possible primes. This gives all indication that if there are [n.sub.i] possible primes in SDS-I, then in SDS-II, the number of possible primes [n.sub.2] = [n.sub.1] * (3/20) * 9/5) = 0.27[n.sub.1] This explains why the number of primes found in SDS-11 is fewer as compared to number of primes found in SDS-I. The time required to search for primes in SDS-I is also correspondingly higher than the time required to search for primes in SDS-II.

Conjecture 2. Every prime divides some element of the sequence. It has been checked that every prime tip to 2591 divides some element of the sequence tip to 10000 terms, so it is again quite reasonable to conjecture that: every prime divides some element of the sequence. Can this be proved?

References

[1] F. Smarandache, Only Problems, Not Solutions, Chicago, Xiquan Publishing House, 1993.

[2] K. Kashihara, Comments and 'Topics on Smarandache Notions and Problems, Erhus University Press, Vail, Arizona, 1996.

[3] Charles Ashbacher., Some Problems Concerning The Smarandache Deconstructive Sequence, Journal of Recreational Mathematics, 2(1998). Bay-Wood Publishing Company, Inc.

[4] Henry Ibstedt, On a concatenation problem, Smarandache Notions Journal, 13(2002), pp. 96-106.

[5] Charles Ashbacher, Pluckings from the tree of Smarandache Sequences and Functions, American Research Press, 1998.

[6] Jason Earls, Some Results Concerning The Smarandache Deconstructive Sequence, Smarandache Notions Journal, 14(2002), pp. 222-226.

[7] Sloane, N.J.A., Sequence A007923, A050234 in" The oil line version of the Encyclopedia of Integer Sequence". http://www.research.att.com/njas/sequences/.

[8] Weisstein, Eric W, "Smarandache Sequences", CRC Concise Encyclopedia is of Mathematics, CRC Press. 1999.

Shyam Sunder Gupta Chief Bridge Engineer, North Central Railway. Allahabad. India
Table 1

Term Prime

2 23
7 4567891
8 23456789
10 1234567891
17 23456789123456789
20 2345678912345789123
25 4567891234567891234567891
28 1234567891234567891234567891
31 7891234567891234567891234567891
38 23456789123456789123456789123456789123
61 4567891234567891234567891234567891234567891234567891234567891
62 23456789123456789123456789123456789123456789123456789123456789
355 789[(123456789).sub.39]1

Table 2

Trailing digits Initial digits

1 1, 4, 7
3 2, 7
6 4, 2
9 2, 1

Table 3

Trailing digits Initial digits

1 1, 4, 7
3 2
9 2

Table 4

Trailing digits Initial digits

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

Table 5

Trailing digits Initial digits

7 1, 6
9 5
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Author:Gupta, Shyam Sunder
Publication:Scientia Magna
Geographic Code:9INDI
Date:Sep 1, 2006
Words:1643
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