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n + k are composites for all nice numbers n, when n+1, 8n+1 both squares

Source: 2022 Saudi Arabia JBMO TST 1.1

November 3, 2022
number theoryPerfect Squares

Problem Statement

The positive n>3n > 3 called ‘nice’ if and only if n+1n +1 and 8n+18n + 1 are both perfect squares. How many positive integers k≤15k \le 15 such that 4n+k4n + k are composites for all nice numbers nn?
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