Problem 1

Let

d

be any positive integer not equal to

2

5

, or

13

. Show that one can find distinct elements

a

and

b

in the set

\{2, 5, 13, d\}

such that

ab - 1

is not a perfect square.

Step 1 of 6: Set up the three squares

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In plain words

If no pairing worked, all three numbers would be squares simultaneously; parity and divisibility constraints on $d$ will then clash.

p^2=2d-1,\quad q^2=5d-1,\quad r^2=13d-1

Detailed analysis

Suppose, for contradiction, that $p^2=2d-1$ , $q^2=5d-1$ , $r^2=13d-1$ for integers $p,q,r$ . We derive a contradiction, which shows at least one of the three cannot be a perfect square.