Ivan Shishkin, Rye (1878)

Problems/Number theoryReviewed

Conjecture de Firoozbakht

by Nono·
100
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
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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Français
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Démontrer que la suite (pnn)n≥1\left(\sqrt[n]{p_{n}}\right)_{n\geq 1} est strictement décroissante. Où pnp_{n} est le n-ième nombre premier.

Une formulation équivalente est de dire que pour tout n≥1n\geq 1
pn+1<pn1+1np_{n+1}<p_{n}^{1+\frac{1}{n}}

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