HMMT 十一月 2021 · 冲刺赛 · 第 5 题

HMMT November 2021 — Guts Round — Problem 5

[6] A perfect power is an integer n that can be represented as a for some positive integers a ≥ 1 and k ≥ 2. Find the sum of all prime numbers 0 < p < 50 such that p is 1 less than a perfect power.

解析

[6] A perfect power is an integer n that can be represented as a for some positive integers a ≥ 1 and k ≥ 2. Find the sum of all prime numbers 0 < p < 50 such that p is 1 less than a perfect power. Proposed by: Joseph Heerens Answer: 41 k k − 1 k − 2 Solution: First, it is known that a − 1 = ( a − 1)( a + a + ... ) . This means either a − 1 or k − 1 k − 2 k a + a + ... + 1 must be 1 in order for a − 1 to be prime. But this only occurs when a is 2. Thus, k the only possible primes are of the form 2 − 1 for some integer k > 1. One can check that the primes 2 3 5 of this form less than 50 are 2 − 1 = 3, 2 − 1 = 7, and 2 − 1 = 31.