A Sierpinski number is an odd natural number k such that is not prime for all natural numbers n. In 1960, Sierpinski proved that there are infinitely many odd integers k with this property, but failed to give an example. Numbers in such a set with odd k and k < 2^n are called Proth numbers.
78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909,….
-List of some known Sierpinski Numbers
The Sierpinski problem asks what the smallest Sierpinski number is. In 1967, Sierpinski and Selfridge conjectured that 78557 is the smallest Sierpinski number. To show this is true, we need to show that all the odd numbers smaller than 78557 are not Sierpinski numbers, i.e. for every odd k below 78557 there is a positive interger n such that is prime. There are only five numbers which have not been eliminated:
k = 21181, 22699, 24737, 55459, and 67607
Find out more here. M x