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min p, smallest prime divisor of n when s=a+b=a^2+b^2=m/n

Source: 2016 Argentina OMA Finals L3 p5

January 16, 2023

number theoryprime divisorsdivisoralgebra

Problem Statement

Let

a

and

b

be rational numbers such that

a+b=a^2+b^2

. Suppose the common value

s=a+b=a^2+b^2

is not an integer, and let's write it as an irreducible fraction:

s=\frac{m}{n}

. Let

p

be the smallest prime divisor of

n

. Find the minimum value of

p