Ivan Shishkin, Rye (1878)

Problems/Number theoryExerciseUnreviewed

Formule de Legendre pour la factorielle

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41
Difficulty scaleÉchelle de difficulté

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.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
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Soient nn un entier strictement positif et pp un nombre premier. Montrer que

vp(n!)=∑i=1+∞⌊npi⌋,v_p(n!)=\sum_{i=1}^{+\infty}\left\lfloor\frac{n}{p^i}\right\rfloor,

où vpv_p désigne la valuation pp-adique et ⌊x⌋\lfloor x\rfloor la partie entière de xx.

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