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 4 of 6: Refine the remaining case modulo 16
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d≡1(mod4) ⟹ d≡1,5,9,13(mod16)d\equiv1\pmod4\ \Longrightarrow\ d\equiv1,5,9,13\pmod{16}
Detailed analysis

Only d≡1(mod4)d\equiv1\pmod4 remains, whose residues modulo 1616 are 1,5,9,131,5,9,13. Perfect squares mod 1616 lie in {0,1,4,9}\{0,1,4,9\}, so we check 13d−113d-1 and 5d−15d-1 in these four subcases.