MathLabs

Problem 2

Determine all positive integers n for which (n^2+1)/(floor(sqrt(n))^2+2) is an integer. Here floor(r) denotes the greatest integer less than or equal to r.
Step 5 of 6: Exclude k=2
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k=2:(a−2)2+1≡1,2,5(mod8),2(m2+2)≡4,6(mod8).k=2:\quad(a-2)^2+1\equiv1,2,5\pmod8,\quad2(m^2+2)\equiv4,6\pmod8.
Detailed analysis

A square modulo 8 is 0, 1, or 4, so the left side can only be 1, 2, or 5 modulo 8. Since m^2 modulo 4 is 0 or 1, the right side is 4 or 6 modulo 8. They cannot be equal.