The number of primes sumni=1(-1){n-i}i! is finite
Abstract
For a positive integer n let $ A_(n+1)=sum _(i=1..^n) (-1)^(n-i) i!, !n=sum _(i=0..n-1) i! and let p_1=3612703. The number of primes of the form A_n is finite, because if n>= p_1, then A_n is divisible by p_1. The heuristic argument is given by which there exists a prime p such that p|!n for all large n; a computer check however shows that this prime has to be greater than 2^{23}. The conjecture that the numbers !n are squarefree is not true because 54503^2|!26541.
- Publication:
-
Mathematics of Computation
- Pub Date:
- 1999
- Bibcode:
- 1999MaCom..68..403Z