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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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


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:
  • Received by editor(s): July 19, 1996
  • Received by editor(s) in revised form: January 23, 1997
  • © Copyright 1999 American Mathematical Society
  • Journal: Math. Comp. 68 (1999), 403-409
  • MSC (1991): Primary 11B83; Secondary 11K31
  • DOI:
  • MathSciNet review: 1484905