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PARITY BIAS IN FUNDAMENTAL UNITS OF REAL QUADRATIC FIELDS
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- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 03 February 2025, pp. 1-4
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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.
ON THE CLASS NUMBER AND THE FUNDAMENTAL UNIT OF THE REAL QUADRATIC FIELD
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 85 / Issue 3 / June 2012
- Published online by Cambridge University Press:
- 25 January 2012, pp. 359-370
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- June 2012
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For a real quadratic field
, let tk be the exact power of 2 dividing the class number hk of k and ηk the fundamental unit of k. The aim of this paper is to study tk and the value of Nk/ℚ(ηk). Various methods have been successfully applied to obtain results related to this topic. The idea of our work is to select a special circular unit ℰk of k and investigate C(k)=〈±ℰk 〉. We examine the indices [E(k):C(k)] and [C(k):CS (k)] , where E(k) is the group of units of k, and CS (k) is that of circular units of k defined by Sinnott. Then by using the Sinnott’s index formula [E(k):CS (k)]=hk, we obtain as much information about tk and Nk/ℚ (ηk) as possible.
, let 