Naslov
Class number one quadratic fields and solvability of some Pellian equations

Autori
Širola, Boris

Izvornik
Acta Mathematica Hungarica (0236-5294) 104 (2004), 1-2; 127-142

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
quadratic field; quadratic form; principal ideal; class number; Pellian equation

Sažetak
Consider these two types of positive square-free integers $d\neq 1$ for which the class number h of the quadratic field $Q(\sqrt (d))$ is odd: (1) d is prime $\not\equiv 1(mod 8)$, or d=2q where q is prime $\equiv 3(mod 4)$, or d=qr where q and r are primes such that $q\equiv 3(mod 8)$ and $r\equiv 7(mod 8)$ ; (2) d is prime $\equiv 1(mod 8)$, or d=qr where q and r are primes such that $q\equiv r\equiv 3 or 7(mod 8)$. For d of type (2) (resp. (1)), let $\Pi$ be the set of all primes (resp. odd primes) $p\in N$ satisfying (d/p)=1. Also, let $\delta =0$ (resp. $\delta =1$) if $d\equiv 2, 3(mod 4)$ (resp. $d\equiv 1(mod 4)$). Then the following are equivalent: (a) h=1 ; (b) For every $p\in\Pi$ at least one of the two Pellian equations $Z^2-dY^2=\pm 4^{;\delta};p$ is solvable in integers ; (c) For every $p\in\Pi$ the Pellian equation $W^2-dV^2=4^{;\delta};p^2$ has a solution (w, v) in integers such that gcd(w, v) divides $2^{;\delta};$.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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