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Problem 574

AMC 12 late, AIME early

Number theory Difficulty 5.0 Find the answer

19.3.13 Find a set of prime numbers $p, q, r$ , such that the numbers $p(p+1), q(q+1), r(r+1)$ form an increasing arithmetic sequence.

Your answer

Official solution

According to the identity
$n(n+1)+(41 n+20)(41 n+21)=2(29 n+14)(29 n+15),$

the numbers $n(n+1),(29 n+14)(29 n+15)$ , and $(41 n+20)(41 n+21)$ form an arithmetic sequence. If for some positive integer $n$ , the numbers $n, 29 n+14, 41 n+20$ are all prime, then they constitute a solution to this problem. The smallest such $n=127, p=127, q=3697, r=5527$ is a solution to this problem.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.