MathLabs

Problem 4

Find all ordered pairs of positive integers (m,n)(m,n) for which n3+1mn−1\frac{n^3+1}{mn-1} is an integer.
Step 4 of 4: Solve the second case and small values
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In plain words

The second case is immediately too large, while n=1,2,3n=1,2,3 can be checked directly.

m+k=2n⟹n+2=m(2n−m)m+k=2n\Longrightarrow n+2=m(2n-m)
Detailed analysis

Here n+2=m(2n−m)n+2=m(2n-m). Taking m≥k=2n−mm\ge k=2n-m, we have k≥2k\ge2 (the case k=1k=1 would give the non-solution m=2n−1m=2n-1), hence the right side is at least 2m≥2n>n+22m\ge2n>n+2. For n=1,2,3n=1,2,3, direct divisor checks give the complete list (1,2),(1,3),(2,1),(2,2),(2,5),(3,1),(3,5),(5,2),(5,3)(1,2),(1,3),(2,1),(2,2),(2,5),(3,1),(3,5),(5,2),(5,3).