David Dummit and Hershy Kisilevsky observed from calculation that the Legendre symbols \( (p/q)\) and \( (q/p)\) are unequal for rather more than a quarter of the pairs of odd primes \( p \) and \( q\) with \( pq\leq x\) , during some calculations. In fact almost \( 30 \% \)
of the \( pq\)'s up to a million satisfy \( p\equiv q\equiv 3 \pmod 4\) . Together we found that this is no accident and that the bias up to \( x\) is roughly \( 1 +1/3(\log\log x-1)\) . This is a much stronger bias than the traditional "prime race" problem. When doing the math one finds that this problems about \( pq\) 's is equivalent to the prime race problem, for primes \( \equiv 3 \pmod 4\) versus those \( \equiv 1 \pmod 4\), in which we weight each prime by its reciprocal.