Problem
NT-B2-M02-P019 Parity of an Exponent
#19
★★★★★ Level 5 of 5
Prove that if \(N=x^2+y^2\), then every prime divisor \(p\equiv3\pmod4\) occurs in the prime factorisation of \(N\) with even exponent.
Use the lemma: such a prime dividing a sum of two squares divides both numbers.
Let \(p\equiv3\pmod4\) and \(p\mid x^2+y^2\). By the lemma, \(p\mid x\) and \(p\mid y\), so \(x=px_1\), \(y=py_1\). Then \(N=p^2(x_1^2+y_1^2)\). If \(p\) again divides \(x_1^2+y_1^2\), repeat the same step. Each step decreases the exponent of \(p\) by \(2\). Therefore the exponent of \(p\) in \(N\) must be even.
This is a strong result, but the proof is accessible after one key lemma.