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A Sierpinski number is a positive, odd integer k for which the integers k.2n+1 are all composite
(that is, for every positive integer n). In 1960 Sierpinski showed that
there were infinitely many such numbers k (all solutions to a family of
congruences), but he did
not explicitly give a numerical example. The congruences provided a sufficient,
but not necessary, condition for an integer to be a Sierpinski number. Therefore
Sierpinski also asked what the smallest such number might be? Determining this
smallest number is now called the Sierpinski problem.
If the congruences proposed by Sierpinski are solved, a 19-digit number
k is obtained as their smallest solution. The much smaller example
k = 78557, now conjectured to be the
smallest Sierpinski number, was found by John Selfridge in 1962. To prove this
conjecture, all you need to do for each of the smaller values of
k, is to find an exponent n which makes
k.2n+1 prime (for that particular value
of k).
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