Lemma 5.7.4 — A prime obstruction to primitive representations

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For every pair of natural numbers $n,p$, assuming $0<n$, that $p$ is prime, that $p$ divides $n$ in the natural numbers, and that the remainder of $p$ on division by $4$ is $3$, there do not exist integers $x,y$ satisfying both $n=x^2+y^2$, with $n$ interpreted as an integer, and $\gcd(x,y)=1$. Here $p\mid n$ means that $n=pm$ for some $m\in\mathbb N$, and primality means that $p\ge2$ and its only natural divisors are $1$ and $p$. The integer gcd used in the statement has a natural-number value and is defined as the natural-number gcd of $|x|$ and $|y|$; it has value $0$ at $(0,0)$, value $|y|$ at $(0,y)$, and value $|x|$ at $(x,0)$. The quantified integers $x,y$ may be negative or zero; the gcd condition is part of the forbidden conjunction, and the statement does not prohibit representations whose gcd is greater than $1$. There is no restriction on the exponent with which $p$ divides $n$. The hypothesis excludes $n=0$; for $n=1$ no $p$ satisfies the prime-divisor hypothesis, so the implication has no such instance. There are no further assumptions.