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


Hi Willaim,

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

m2 - (n + 1)m + (n2 - n) = 0

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

(n + 1)2 -4(n2 -n)

must be a square. Simplified this expression is

-3n2 + 6n + 1

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.

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