Subject: Calculus Problem

Hi my name is William and I'm a 12th grade student pondered by a problem my calculus teacher gave me.

Find all pair of positive integers m and n such that mn + m + n divides m² + n² = 1.

Please I would appreciate any type of help

William

If (mn + m + n)/(m² + n²) = 1 then mn + m + n = m² + n². Rearrange the terms to write this expression as a quadratic in m, that is

If you use the general quadratic to solve for m, and m is to be an integer, then the discriminant (the piece under the root sign) must be a square. That is

must be a square. Simplified this expression is

Notic that when n is large this expression is negative and hence can't be a square. Since n is a positive integer substitute n = 1, 2, 3,... until the expression is negative. For each positive value that is a square substitute the value of n into the original expression and solve for m.