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Sierpiński's Composite Number Theorem

As proved by Sierpiński (1960), there exist infinitely many positive odd numbers k such that k·2^n+1 is composite for every n>=1. Numbers k with this property are called Sierpiński numbers of the second kind, and analogous numbers with the plus sign replaced by a minus are called Riesel numbers. It is conjectured that the smallest value of k for a Sierpiński number of the second kind is k=78557 (although a handful of smaller candidates remain to be eliminated) and that the smallest Riesel number is k=509203.

See also

Cunningham Number, Proth Number, Proth Prime, Riesel Number, Sierpiński Number of the Second Kind, Sierpiński's Prime Sequence Theorem

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References

Ballinger, R. "The Riesel Problem: Definition and Status." http://www.prothsearch.net/rieselprob.html.Ballinger, R. "The Sierpinski Problem: Definition and Status." http://www.prothsearch.net/sierp.html.Ballinger, R. and Keller, W. "The Riesel Problem: Search for Remaining Candidates." http://www.prothsearch.net/rieselsearch.html.Buell, D. A. and Young, J. "Some Large Primes and the Sierpiński Problem." SRC Tech. Rep. 88004, Supercomputing Research Center, Lanham, MD, 1988.Helm, L. and Norris, D. "Seventeen or Bust: A Distributed Attack on the Sierpinski Problem." http://www.seventeenorbust.com/.Jaeschke, G. "On the Smallest k such that k·2^N+1 are Composite." Math. Comput. 40, 381-384, 1983.Jaeschke, G. Corrigendum to "On the Smallest k such that k·2^N+1 are Composite." Math. Comput. 45, 637, 1985.Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k·2^n+1." Math. Comput. 41, 661-673, 1983.Keller, W. "Factors of Fermat Numbers and Large Primes of the Form k·2^n+1, II." Preprint available at http://www.rrz.uni-hamburg.de/RRZ/W.Keller/.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 357-359, 1996.Riesel, H. "Några stora primtal." Elementa 39, 258-260, 1956.Sierpiński, W. "Sur un problème concernant les nombres k·2^n+1." Elem. d. Math. 15, 73-74, 1960.

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Sierpiński's Composite Number Theorem

Cite this as:

Weisstein, Eric W. "Sierpiński's Composite Number Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SierpinskisCompositeNumberTheorem.html

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