A proof of Sondow's conjecture on the Smarandache function

2019/06/30 by Xiumei Li, Min Sha, Li, Xiumei +1
Mathematics · #11A25 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A25

paper · pdf · doi:10.48550/arxiv.1907.00370

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arxiv created 2020/02/07 · arxiv updated 2020/02/11

Abstract

The Smarandache function of a positive integer n, denoted by S(n), is defined to be the smallest positive integer j such that n divides the factorial j!. In this note, we prove that for any fixed number k > 1, the inequality n^k < S(n)! holds for almost all positive integers n. This confirms Sondow's conjecture which asserts that the inequality n² < S(n)! holds for almost all positive integers n.

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