The number of primes is finite

Author:
Miodrag Živković

Journal:
Math. Comp. **68** (1999), 403-409

MSC (1991):
Primary 11B83; Secondary 11K31

DOI:
https://doi.org/10.1090/S0025-5718-99-00990-4

MathSciNet review:
1484905

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Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Miodrag Živković**

Affiliation:
Matematički Fakultet, Beograd

Email:
ezivkovm@matf.bg.ac.yu

Keywords:
Prime numbers,
left factorial,
divisibility

Received by editor(s):
July 19, 1996

Received by editor(s) in revised form:
January 23, 1997

Article copyright:
© Copyright 1999
American Mathematical Society