Sierpiński's Composite Number Theorem
As proved by Sierpiński (1960), there exist infinitely many positive odd numbers 
such that 
is composite
for every 
. Numbers
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 
for a Sierpiński
number of the second kind is 
(although
a handful of smaller candidates remain to be eliminated) and that the smallest Riesel number is 
.
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such that 
are Composite." Math.
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such that 
are Composite." Math.
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."
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, II."
Preprint available at https://www.rrz.uni-hamburg.de/RRZ/W.Keller/.
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."
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