Sufficient condition for non-congruent $pq$ and $2pq$ with primes $p$ and $q$ in the form of $8k+1$
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Within this article, our focus lies on positive integers represented as $pq$ and $2pq$, where $p$ and $q$ are prime numbers adhering to the conditions $p\equiv q \equiv 1 \pmod{8}$. Additionally, we will operate under the assumption that $q$ is a quadratic non-residue modulo $p$. Our aim is to present a condition that ensures that $n$ is a non-congruent number, and we will substantiate the sufficiency of this condition through proof.
Abstract
Within this article, our focus lies on positive integers represented as $pq$ and $2pq$, where $p$ and $q$ are prime numbers adhering to the conditions $p\equiv q \equiv 1 \pmod{8}$. Additionally, we will operate under the assumption that $q$ is a quadratic non-residue modulo $p$. Our aim is to present a condition that ensures that $n$ is a non-congruent number, and we will substantiate the sufficiency of this condition through proof.