# Search Results

## You are looking at 1 - 1 of 1 items for

- Author or Editor: Boris Širola x

- All content x

## 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 *q*≡*r*≡ 3 or 7 (mod 8). For *d* of type (2) (resp. (1)), let Π be the set of all primes (resp. odd primes) *p*∈**N** 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} = 4^{δ}
*p* is solvable in integers. (c) For every p∈П the Pellian equation *W*
^{2}-*dV*
^{2} = 4^{δ}
*p*
^{2} has a solution (*w,v*) in integers such that gcd (*w,v*) divides 2^{δ}.