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On the non-existence of abelian conditions governing solvability of the - 1 Pell equation. (English) Zbl 0689.12006

It is shown that the infinite sequence of fundamental units \(\{\epsilon_{p_ 1p_ 2...p_ n},n\geq 1\}\) in the fields \({\mathbb{Q}}(\sqrt{p_ 1p_ 2...p_ n})\) can have arbitrary sequence of signatures; that is, for any sequence of signs \(\{\sigma_ n\}\), with \(\sigma_ 1=-1\), there are finitely many sequences of primes \(\{p_ n\}\) for which \(Norm \epsilon_{p_ 1p_ 2...p_ n}=\sigma_ n.\) The primes can be taken from arbitrary reduced residue classes for a given sequence of moduli, if the set of prime divisors of the moduli has Dirichlet density 0. This shows that simple congruence conditions do not suffice to determine solvability of the equation \(x^ 2-dy^ 2=-1\), if the moduli for these congruences are relatively prime to d. Along the same lines, it is shown that there are no ”Abelian” conditions which can characterize the solvability of ths equation for all d; these are conditions which can be expressed in terms of the splitting of one of the prime divisors of d in an Abelian extension of \({\mathbb{Q}}\).
Reviewer: P.Morton

MSC:

11R23 Iwasawa theory
11D04 Linear Diophantine equations
11R18 Cyclotomic extensions
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