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 .
See also Cunningham Number, Sierpinski Number of the Second Kind
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.
See also Composite Number, Sierpinski Numbers of the Second Kind, Sierpinski's Prime Sequence Theorem