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Strange inequality on primes and factorization...

Source: Italian TST , day 2, n°3

June 2, 2007
inequalitiesgeometrygeometric transformationreflectionnumber theoryprime numbersnumber theory proposed

Problem Statement

Let p≥5p \geq 5 be a prime.
(a) Show that exists a prime q≠pq \neq p such that q∣(p−1)p+1q| (p-1)^{p}+1
(b) Factoring in prime numbers (p−1)p+1=∏i=1npiai(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}} show that: ∑i=1npiai≥p22\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2
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