The number of primes is finite

Živković, Miodrag

doi:10.1090/S0025-5718-99-00990-4

The number of primes is finite
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Math. Comp. 68 (1999), 403-409 Request permission

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 $n\geq p_1$, then $A_n$ is divisible by $p_1$. The heuristic argument is given by which there exists a prime $p$ such that $p\mid !n$ 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 ${54503^2}\mid {!26541}$.

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