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PARITY BIAS IN FUNDAMENTAL UNITS OF REAL QUADRATIC FIELDS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 03 February 2025, pp. 1-4
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- Article
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We compute primes
$p \equiv 5 \bmod 8$ up to 
$10^{11}$ for which the Pellian equation 
$x^2-py^2=-4$ has no solutions in odd integers; these are the members of sequence A130229 in the Online Encyclopedia of Integer Sequences. We find that the number of such primes 
$p\leqslant x$ is well approximated by 
$$ \begin{align*}\frac{1}{12}\pi(x) - 0.037\int_2^x \frac{dt}{t^{1/6}\log t},\end{align*} $$where
$\pi (x)$ is the usual prime counting function. The second term shows a surprising bias away from membership of this sequence.
