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Necessary and sufficient conditions for the class number of a real quadratic field to be one, and a conjecture of S. Chowla. (English) Zbl 0673.12005
Subject of the paper is a conjecture of Chowla that the class number h(n) of a real quadratic number field \(K={\mathbb{Q}}(\sqrt{n})\) is greater than 1 where the radicant n is a prime number of the form \(n=m^ 2+1\), \(m>26\). Using elementary techniques the author is able to prove this conjecture for all such radicants except for those with \(m=2q\), q a prime number. - By using different techniques similar results were obtained by F. Callialp [C. R. Acad. Sci., Paris, Sér. A 291, 623-625 (1980; Zbl 0499.12007)].
For the remaining case, the author shows that \(h(4q^ 2+1)=1\) is tantamount to \(-x^ 2+x+q^ 2\) being prime for all integers x with \(1<x<q\).
Reviewer: J.Buchmann

MSC:
11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11N32 Primes represented by polynomials; other multiplicative structures of polynomial values
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