Class number one criteria for real quadratic fields with discriminant \(k^2p^2\pm 4p\) (Q2869642)
From MaRDI portal
 | This is the item page for this Wikibase entity, intended for internal use and editing purposes. Please use this page instead for the normal view: Class number one criteria for real quadratic fields with discriminant \(k^2p^2\pm 4p\) |
scientific article; zbMATH DE number 6242866
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Class number one criteria for real quadratic fields with discriminant \(k^2p^2\pm 4p\) |
scientific article; zbMATH DE number 6242866 |
Statements
3 January 2014
0 references
real quadratic field
0 references
class number one criteria
0 references
prime-producing quadratic polynomial
0 references
Class number one criteria for real quadratic fields with discriminant \(k^2p^2\pm 4p\) (English)
0 references
Let \(\Delta= 1+4m\) be the discriminant of the real quadratic field \(K= \mathbb{Q}(\sqrt{\Delta})\), and let \(h_K\) be the class number of \(K\). In [Int. J. Math. Math. Sci. 2009, Article ID 819068, 14 p. (2009; Zbl 1290.11151)], the first author conjectured that the following are equivalent: (i) \(\Delta=pq\), where \(p<q\) are primes and \(|x^2+ x-m|\) is a prime for all \(x\in [(p+1)/2, \sqrt{m}+(p- 1)/2]\). (ii) \(h_K= 1\) and \(\Delta= 9p^2\pm 4p\). Recently, the authors [Funct. Approximatio, Comment. Math. 45, No. 2, 271--288 (2011; Zbl 1296.11138)] verified the above conjecture.NEWLINENEWLINE In the present paper the authors give a full generalization for \(\Delta= k^2p^2\pm 4p\), where \(k\) is an odd positive integer.
0 references
