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 3 of 6: Limit the possible quotient
Detailed analysis
From 0≤a≤2m, the numerator is at most max{4,(2m−2)^2}+1≤4m^2+1<4(m^2+2). It is positive, so its quotient by m^2+2 must be 1, 2, or 3.