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Clear All Modify Search  # Class number one quadratic fields and solvability of some Pellian equations

Acta Mathematica Hungarica
Author: Boris Širola

## Abstract

Consider these two types of positive square-free integers d≠ 1 for which the class number h of the quadratic field Q(√d) is odd: (1) d is prime∈ 1(mod 8), or d=2q where q is prime ≡ 3 (mod 4), or d=qr where q and r are primes such that q≡ 3 (mod 8) and r≡ 7 (mod 8); (2) d is prime ≡ 1 (mod 8), or d=qr where q and r are primes such that qr≡ 3 or 7 (mod 8). For d of type (2) (resp. (1)), let Π be the set of all primes (resp. odd primes) pN satisfying (d/p) = 1. Also, let δ :=0 (resp. δ :=1) if d≡ 2,3 (mod 4) (resp. d≡ 1 (mod 4)). Then the following are equivalent: (a) h=1; (b) For every p∈П at least one of the two Pellian equations Z 2-dY 2 = 4δ p is solvable in integers. (c) For every p∈П the Pellian equation W 2-dV 2 = 4δ p 2 has a solution (w,v) in integers such that gcd (w,v) divides 2δ.

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