Problem 4

Find all ordered pairs of positive integers

(m,n)

for which

\frac{n^3+1}{mn-1}

is an integer.

Step 3 of 4: Solve the first case

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In plain words

The equation forces one factor to be within two of $n$ .

m+k=n\Longrightarrow n=m(n-m)-1

Detailed analysis

With $k=n-m$ , the identity becomes $n=m(n-m)-1$ . By symmetry take $m\ge k$ . The value $m=n-1$ fails; if $m=n-2$ , then $n=5$ , giving $(m,n)=(2,5),(3,5)$ . If $m<n-2$ , then $n-m\ge3$ and $m\ge n/2$ , so $m(n-m)-1\ge3n/2-1>n$ , impossible.