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 3 of 6: Handle d≡2(mod4)d\equiv2\pmod4
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2d−1≡3(mod4)2d-1\equiv3\pmod4
Detailed analysis

If d≡2(mod4)d\equiv2\pmod4 then 2d≡0(mod4)2d\equiv0\pmod4, so 2d−1≡3(mod4)2d-1\equiv3\pmod4, which is never the residue 33 modulo 44. Hence 2d−12d-1 is not a perfect square, and we may take (a,b)=(2,d)(a,b)=(2,d).