Big biases amongst products of two primes

TL;DR

This paper investigates the distribution of odd integers that are products of two primes, revealing a significant bias towards primes congruent to 3 modulo 4 in such products.

Contribution

It demonstrates that more than a quarter of these products up to a certain limit have both primes congruent to 3 modulo 4, highlighting a notable bias in their distribution.

Findings

Over 25% of odd semiprimes up to x have both primes ≡ 3 mod 4

Substantial bias towards primes ≡ 3 mod 4 in products of two primes

Quantitative analysis of prime congruence distribution in semiprimes

Abstract

We show that substantially more than a quarter of the odd integers of the form $pq$ up to $x$, with $p,q$ both prime, satisfy $p\equiv q\equiv 3 \pmod 4$.