The number of primes <IMG ALIGN=MIDDLE HEIGHT=34 WIDTH=128> is finite期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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The number of primes is finite

Authors:	Miodrag Zivkovic

Institution:	Matematicki Fakultet, Beograd

Abstract:	For a positive integer let $A_{n+1}=\sum _{i=1}^n (-1)^{n-i} i!,$ $\,!n=\sum _{i=0}^{n-1} i!$ and let . The number of primes of the form is finite, because if $n\geq p_1$ , then is divisible by . The heuristic argument is given by which there exists a prime such that $p\,\vert\,\,!n$ for all large ; 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 $54503^2\,\vert\,\,!26541$ .

Keywords:	Prime numbers left factorial divisibility

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