## 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

Zbl 0499.12007
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### References:

 [1] R. G. Ayoub and S. Chowla, On Euler’s polynomial, J. Number Theory 13 (1981), no. 4, 443 – 445. · Zbl 0472.10009 [2] Fethi Çallialp, Non-nullité des fonctions zéta des corps quadratiques réels pour 0<\?<1, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 12, A623 – A625 (French, with English summary). · Zbl 0499.12007 [3] S. Chowla and J. Friedlander, Class numbers and quadratic residues, Glasgow Math. J. 17 (1976), no. 1, 47 – 52. · Zbl 0323.12006 [4] S. Chowla and J. B. Friedlander, Some remarks on \?-functions and class numbers, Acta Arith. 28 (1975/76), no. 4, 413 – 417. · Zbl 0331.12007 [5] Günter Degert, Über die Bestimmung der Grundeinheit gewisser reell-quadratischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg 22 (1958), 92 – 97 (German). · Zbl 0079.05803 [6] Hong Wen Lu, On the real quadratic fields of class-number one, Sci. Sinica 24 (1981), no. 10, 1352 – 1357. · Zbl 0495.12004 [7] Masakazu Kutsuna, On a criterion for the class number of a quadratic number field to be one, Nagoya Math. J. 79 (1980), 123 – 129. · Zbl 0447.12006 [8] D. H. Lehmer, On the function $${x^2} + x + A$$, Sphinx 6 (1936), 212-214. · JFM 62.1130.01 [9] R. A. Mollin, On the insolubility of a class of Diophantine equations and the nontriviality of the class numbers of related real quadratic fields of Richaud-Degert type, Nagoya Math. J. 105 (1987), 39 – 47. · Zbl 0591.12005 [10] R. A. Mollin, Diophantine equations and class numbers, J. Number Theory 24 (1986), no. 1, 7 – 19. · Zbl 0591.12006 [11] R. A. Mollin, Lower bounds for class numbers of real quadratic fields, Proc. Amer. Math. Soc. 96 (1986), no. 4, 545 – 550. · Zbl 0591.12007 [13] G. Rabinovitch, Eindeutigkeit der Zerlegung in Primzahlfaktoren in Quadratischen Zahlkörpern, J. Reine Angew. Math. 142 (1913), 153-164. · JFM 44.0243.03 [14] C. Richaud, Sur la résolution des équations $${x^2} - A{y^2} = \pm 1$$, Atti Accad. Pontif. Nuovi Lincei (1866), 177-182. [15] H. M. Stark, A complete determination of the complex quadratic fields of class-number one, Michigan Math. J. 14 (1967), 1 – 27. · Zbl 0148.27802
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