A123159 Conjectured smallest Sierpiński numbers of the second kind S, base b=2,3,4,5,..., where S*b^n+1 is composite for all n>=1 and gcd(S+1, b-1) = 1.

78557, 125050976086, 66741, 159986, 174308, 1112646039348, 1, 2344, 9175, 1490, 521, 132, 4, 91218919470156, 2500, 278, 398, 765174, 8, 1002, 6694, 182, 30651, 262638, 221, 8, 4554, 4, 867, 6360528, 1, 1854, 6, 214018, 1886, 2604, 14, 166134, 826477, 8, 13372, 2256, 4, 53474

Offset: 2

Author

Robert Smith (robert_smith44(AT)hotmail.com), Oct 02 2006

Keywords

• nonn

Comments

Some values with base b=2^x+1 for integers x have also been calculated - see the links.

Examples

			For base=3, S+1 should be coprime to 3-1 = 2, so S must be even. Find a covering set of multiplicative orders of primes base b and discover S by trial and error using the Chinese Remainder Theorem.
		

Links

• G. Jaeschke, On the smallest k such that all k * 2^n +1 are composite, Math. Comp., 40:181 (1983) 381-384. MR 84k:10006.
• Primeform Group, base=2^x+1.
• Primeform Group, base=3.
• Primeform Group, base=5.
• Carlos Rivera, Problem 36. The Liskovets-Gallot numbers, The Prime Puzzles and Problems Connection.
• Guido Smetrijns, Andrew J. Walker, Shane Findley, Jim Fougeron, Mikael Klasson, Robert Smith and others, Riesel/Sierpinski in base 3, digest of 43 messages in primeform Yahoo group, May 24, 2004 - Jan 7, 2007. [Cached copy]
• Robert Smith, David Broadhurst, Shane Findley, Sierpinski / Riesel base 2^x+1, digest of 4 messages in primeform Yahoo group, Sep 26 - Sep 27, 2004. [Cached copy]
• Robert Smith, Guido Smetrijns, Mikael Klasson, Riesel Sierpinski in base 5, digest of 6 messages in primeform Yahoo group, Sep 17 - Sep 18, 2004

Crossrefs

Cf. A076336.

Extensions

a(6) from Arkadiusz Wesolowski, Nov 20 2014

a(3) corrected and a(7)-a(45) from Eric Chen, Dec 16 2014