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 5 of 6: Force dd to be even
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4∣(m+n)(m−n) ⟹ 2∣d4\mid(m+n)(m-n)\ \Longrightarrow\ 2\mid d
Detailed analysis

Since m2−n2=2dm^2-n^2=2d is even, mm and nn have the same parity, so both m−nm-n and m+nm+n are even; hence 4∣(m+n)(m−n)=2d4\mid(m+n)(m-n)=2d, forcing dd to be even.