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Number Theory

Source: 2016 China Second Round Olympiad Problem 4

January 8, 2017
Sequencenumber theory

Problem Statement

Let pp and p+2p+2 be primes, p>3p>3. Sequence {an}:a1=2,an=an−1+⌈pan−1n⌉\{a_n\}:a_1=2,a_n=a_{n-1}+\left\lceil{\frac{pa_{n-1}}{n}}\right\rceil. Prove that n∣pan−1+1n\mid pa_{n-1}+1 for all n=3,4,…,p−1n=3,4,\dots,p-1.
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