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A076336 (Provable) Sierpiński numbers: odd numbers n such that for all k >= 1 the numbers n*2^k + 1 are composite. 70
78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, 3083723, 3098059, 3555593, 3608251 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
It is only a conjecture that this sequence is complete up to 3000000 - there may be missing terms.
It is conjectured that 78557 is the smallest Sierpiński number. - T. D. Noe, Oct 31 2003
Sierpiński numbers are proved by exhibiting a periodic sequence p of prime divisors with p(k) | n*2^k+1 and disproved by finding prime n*2^k+1. It is conjectured that numbers that cannot be proved Sierpiński in this way are non-Sierpiński. However, some numbers resist both proof and disproof. - David W. Wilson, Jan 17 2005
Sierpiński showed that this sequence is infinite.
There are 4 related sequences that arise in this context:
S1: Numbers n such that n*2^k + 1 is composite for all k (this sequence)
S2: Odd numbers n such that 2^k + n is composite for all k (apparently it is conjectured that S1 and S2 are the same sequence)
S3: Numbers n such that n*2^k + 1 is prime for all k (empty)
S4: Numbers n such that 2^k + n is prime for all k (empty)
The following argument, due to Michael Reid, attempts to show that S3 and S4 are empty: If p is a prime divisor of n + 1, then for k = p - 1, the term (either n*2^k + 1 or 2^k + n) is a multiple of p (and also > p, so not prime). [However, David McAfferty points that for the case S3, this argument fails if p is of the form 2^m-1. So it may only be a conjecture that the set S3 is empty. - N. J. A. Sloane, Jun 27 2021]
a(1) = 78557 is also the smallest odd n for which either n^p*2^k + 1 or n^p + 2^k is composite for every k > 0 and every prime p greater than 3. - Arkadiusz Wesolowski, Oct 12 2015
n = 4008735125781478102999926000625 = (A213353(1))^4 is in this sequence but is thought not to satisfy the conjecture mentioned by David W. Wilson above. For this multiplier, all n*2^(4m + 2) + 1 are composite by an Aurifeuillean factorization. Only the remaining cases, n*2^k + 1 where k is not 2 modulo 4, are covered by a finite set of primes (namely {3, 17, 97, 241, 257, 673}). See Izotov link for details (although with another prime set). - Jeppe Stig Nielsen, Apr 14 2018
Conjecture: if S is a (provable) Sierpiński number, then there exists a prime P such that S^p is also a Sierpiński number for every prime p > P. - Thomas Ordowski, Jul 12 2022
Problem: are there odd numbers K such that K - 2^m is a Sierpiński number for every 1 < 2^m < K? If so, then all positive values of (K - 2^m)*2^n + 1 are composite. Also, by the dual Sierpiński conjecture, K - 2^m + 2^n is composite for every 1 < 2^m < K and for every n > 0. Note that, by the dual Sierpiński conjecture, if p is an odd prime and 1 < 2^m < p, then there exists n such that (p - 2^m)*2^n + 1 is prime. So if such a number K exists, it must be composite. - Thomas Ordowski, Jul 20 2022
From M. F. Hasler, Jul 23 2022: (Start)
1) The above Conjecture is true for Sierpiński numbers provable by a "covering set", with P equal to the largest prime factor of the elements of that set*, according to the explanation from Michael Filaseta posted Jul 12 2022 on the SeqFan mailing list, cf. links. (*More generally: for S^p with any p coprime to all elements of the covering set, but not necessarily prime.)
2) Wilson's comment from 2005 (also the first part, not only the conjecture) is misleading if not wrong because there are provable Sierpiński numbers for which a covering set is not known (maybe even believed not to exist), as explained by Nielsen in his above comment from 2018. It should be clarified that (or: whether) "provable" in the definition of this sequence does not only mean "through a covering set". (End)
REFERENCES
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 420.
P. Ribenboim, The Book of Prime Number Records, 2nd. ed., 1989, p. 282.
LINKS
T. D. Noe and Arkadiusz Wesolowski, Table of n, a(n) for n = 1..15000 (T. D. Noe supplied 13394 terms which came from McLean. a(1064), a(7053), and a(13397)-a(15000) from Arkadiusz Wesolowski.)
Chris Caldwell, Riesel number
Chris Caldwell, Sierpinski number
Michael Filaseta, quoted by T. Ordowski, Re: Is it true? SeqFan mailing list, Jul 12 2022
Michael Filaseta, Jacob Juillerat, and Thomas Luckner, Consecutive primes which are widely digitally delicate and Brier numbers, arXiv:2209.10646 [math.NT], 2022.
Dan Ismailescu and Peter Seho Park, On Pairwise Intersections of the Fibonacci, Sierpiński, and Riesel Sequences, Journal of Integer Sequences, 16 (2013), #13.9.8.
Anatoly S. Izotov, A Note on Sierpinski Numbers, Fibonacci Quarterly (1995), pp. 206-207.
G. Jaeschke, On the Smallest k Such that All k*2^N + 1 are Composite, Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 381-384.
J. McLean, Brier Numbers [Cached copy]
Carlos Rivera, Problem 29. Brier numbers, The Prime Puzzles and Problems Connection.
Payam Samidoost, Dual Sierpinski problem search page [Broken link?]
Payam Samidoost, Dual Sierpinski problem search page [Cached copy]
Payam Samidoost, 4847 [Broken link?]
Payam Samidoost, 4847 [Cached copy]
W. Sierpiński, Sur un problème concernant les nombres k * 2^n + 1, Elem. Math., 15 (1960), pp. 73-74.
Jeremiah T. Southwick, Two Inquiries Related to the Digits of Prime Numbers, Ph. D. Dissertation, University of South Carolina (2020).
Eric Weisstein's World of Mathematics, Sierpiński Number of the Second Kind
Wikipedia, Sierpiński number.
CROSSREFS
Sequence in context: A249084 A038826 A038815 * A244562 A368560 A123159
KEYWORD
nonn,hard,nice
AUTHOR
N. J. A. Sloane, Nov 07 2002
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)