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: Combine into a difference of squares
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r2−q2=8d ⟹ (m+n)(m−n)=2dr^2-q^2=8d\ \Longrightarrow\ (m+n)(m-n)=2d
Detailed analysis

Since q=2n,r=2mq=2n,r=2m, we get r2−q2=(13d−1)−(5d−1)=8dr^2-q^2=(13d-1)-(5d-1)=8d, i.e. 4m2−4n2=8d4m^2-4n^2=8d, so m2−n2=(m+n)(m−n)=2dm^2-n^2=(m+n)(m-n)=2d.