Today's number is 78,557
A Sierpiński number is an odd natural number k such that k * 2^n + 1 is composite for all natural numbers n.
The sequence of currently known Sierpiński numbers is:
78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, . . .
This is sequence A076336 in the OEIS.
78,557 is the first number in this list. This number earned its Sierpiński badge thanks to John Selfridge who in 1962 showed that all numbers of the form 78,557 * 2^n + 1 have one of these factors: 3, 5, 7, 13, 19, 37, 73
There is also something known as the Sierpiński problem, which asks for the value of the smallest Sierpiński number. Selfridge, in private correspondence with Paul Erdős, guessed that 78,557 holds the crown.
To prove this, one must show that all of the odd numbers smaller than 78,557 are *not* Sierpiński numbers. That is, mathematicians must show that every odd number smaller than 78,557 eventually produces a prime when plugged into k * 2^n + 1. That's a lot of calculations, but luckily there is a distributed volunteer computing project known as PrimeGrid that is currently going through the remaining k values, which I've listed below:
k = 21181, 22699, 24737, 55459, and 67607.
So the next time you think obedient little creatures, remember 78,557: a number that refuses to cooperate with primes.
