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13^k=a^2 +b^2

Source: China Northern MO 2013 p4 CNMO

November 25, 2022

number theorySum of Squares

Problem Statement

For positive integers

n,a,b

, if

n=a^2 +b^2

, and

a

and

b

are coprime, then the number pair

(a,b)

is called a square split of

n

(the order of

a, b

does not count). Prove that for any positive

k

, there are only two square splits of the integer

13^k

.

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