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: qq and rr must both be even
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q2=2(5k2−5k+2),r2=2(13k2−13k+6)q^2=2(5k^2-5k+2),\quad r^2=2(13k^2-13k+6)
Detailed analysis

Substituting d=2k2−2k+1d=2k^2-2k+1 into q2=5d−1q^2=5d-1 and r2=13d−1r^2=13d-1 shows both expressions are even; hence q,rq,r are even, and we write q=2nq=2n, r=2mr=2m.