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The number of primes $ \sum _{i=1}^{n} (-1)^{n-i}i!$ is finite

Author: Miodrag Zivkovic
Journal: Math. Comp. 68 (1999), 403-409
MSC (1991): Primary 11B83; Secondary 11K31
MathSciNet review: 1484905
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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\,\vert\,\,!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\,\vert\,\,!26541$.

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Additional Information

Miodrag Zivkovic
Affiliation: Matematički Fakultet, Beograd

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