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 4 of 6: Exclude k=1
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k=1⟹(a−2)2−m2=1⟹m=0, impossible.k=1\Longrightarrow(a-2)^2-m^2=1\Longrightarrow m=0,\text{ impossible}.
Detailed analysis

If k=1, then (a−2)^2−m^2=1. The difference of the two squares can equal 1 only with m=0 and |a−2|=1, contradicting m≥1.