Sierpinski's Composite Number Theorem

There exist infinitely many Odd Integers $k$ such that $k\cdot 2^n+1$ is Composite for every $n\geq 1$. Numbers $k$ 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 $k=78{,}557$ and the smallest Riesel Number is $k=509{,}203$.

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 $k$ such that $k\cdot 2^N+1$ are Composite.'' Math. Comput. 40, 381-384, 1983.

Jaeschke, G. Corrigendum to ``On the Smallest $k$ such that $k\cdot 2^N+1$ are Composite.'' Math. Comput. 45, 637, 1985.

Keller, W. ``Factors of Fermat Numbers and Large Primes of the Form $k\cdot 2^n+1$.'' Math. Comput. 41, 661-673, 1983.

Keller, W. ``Factors of Fermat Numbers and Large Primes of the Form $k\cdot 2^n+1$, 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 $k\cdot 2^n+1$.'' Elem. d. Math. 15, 73-74, 1960.

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