Sierpinski's Composite Number Theorem

There exist infinitely many Odd Integers such that is Composite for every. Numbers with this property are called Sierpinski 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 Sierpinski Number of the Second Kind is and the smallest Riesel Number is .

References

Buell, D. A. and Young, J. ``Some Large Primes and the Sierpinski Problem.'' SRC Tech. Rep. 88004, Supercomputing Research Center, Lanham, MD, 1988.

Jaeschke, G. ``On the Smallest such that are Composite.'' Math. Comput. 40, 381-384, 1983.

Jaeschke, G. Corrigendum to ``On the Smallest such that are Composite.'' Math. Comput. 45, 637, 1985.

Keller, W. ``Factors of Fermat Numbers and Large Primes of the Form .'' Math. Comput. 41, 661-673, 1983.

Keller, W. ``Factors of Fermat Numbers and Large Primes of the Form , II.'' In prep.

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.

Sierpinski, W. ``Sur un problème concernant les nombres .'' Elem. d. Math. 15, 73-74, 1960.

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