MathLabs

Problem 1

Let dd be any positive integer not equal to 22, 55, or 1313. Show that one can find distinct elements aa and bb in the set {2,5,13,d}\{2, 5, 13, d\} such that ab−1ab - 1 is not a perfect square.
Step 2 of 6: pp is odd, so dd has a fixed odd form
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p=2k−1 ⟹ d=2k2−2k+1p=2k-1\ \Longrightarrow\ d=2k^2-2k+1
Detailed analysis

Since 2d−12d-1 is odd, pp is odd; write p=2k−1p=2k-1. Substituting into p2=2d−1p^2=2d-1 gives d=2k2−2k+1d=2k^2-2k+1, which is odd for every integer kk.