Ivan Shishkin, Rye (1878)

Problems/Number theoryExerciseUnreviewed

Legendre formula for the factorial

by Sequoia·
41
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This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

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Let nn be a positive integer and pp be a prime number. Show that
vp(n!)=∑i=1+∞⌊npi⌋,v_p(n!)=\sum_{i=1}^{+\infty}\left\lfloor \frac{n}{p^i}\right\rfloor,where vpv_p is the pp-adic valuation and ⌊x⌋\lfloor x\rfloor the floor function evaluated at xx.

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