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30 divides n_1^4 + n_2^4+...+n_{31}^4, among 31 primes exist 3 consecutive

Source: 2003 Romania JBMO TST 1.2

June 1, 2020
primesnumber theoryconsecutivedividesSum of powers

Problem Statement

Consider the prime numbers n1<n2<...<n31n_1< n_2 <...< n_{31}. Prove that if 3030 divides n14+n24+...+n314n_1^4 + n_2^4+...+n_{31}^4, then among these numbers one can find three consecutive primes.