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When is 133...3 prime?

Source: Germany 2024, Problem 4

June 14, 2024
number theorynumber theory proposed

Problem Statement

Let k>2k>2 be a positive integer such that the kk-digit number nk=133…3n_k=133\dots 3, consisting of a digit 11 followed by k−1k-1 digits 33 is prime. Show that 24∣k(k+2)24 \mid k(k+2).
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