Class number one problem for the real quadratic fields $${{\mathbb {Q}({\sqrt{m^2+2r}})}}$$ Q
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Archiv der Mathematik
Class number one problem √ for the real quadratic fields Q( m2 + 2r) Azizul Hoque and Srinivas Kotyada
Abstract. We investigate the class number √ one problem for a parametric family of real quadratic fields of the form Q( m2 + 4r) for certain positive integers m and r. Mathematics Subject Classification. Primary 11R29; Secondary 11R11. Keywords. Class number one problem, Real quadratic field, Pell-type equation.
1. Introduction. A well-known conjecture of Gauss states that there are infinitely many real quadratic fields with class number one. This is still unresolved. Attempts to prove the conjecture have led to important ideas that were instrumental in throwing light on some particular types of quadratic fields, for example for the so-called extended √ Richaud-Degert type real quadratic fields. Recall that a real quadratic field Q( d) is of extended Richaud-Degert type if d is a square-free positive integer of the form m2 +r with r | 4m and −m < r ≤ m or r = ±4m/3. The work of Louboutin [7], Mollin and Williams [8], and Yokoi √ [10] confirmed that there are 43 real quadratic fields Q( d), d = 2, 3, 5, 6, 7, 11, 13, 14, 17, 21, 23, 29, 33, 37, 38, 47, 53, 62, 69, 77, 83, 93, 101, 141, 167, 173, 197, 213, 227, 237, 293, 398, 413, 437, 453, 573, 677, 717, 1077, 1133, 1253, 1293, 1757, of extended Richaud-Degert type with class number one with possibly one more such field. However, under the generalized Riemann hypothesis, there exists at most one more such field with class number one. The problem of finding this exceptional real quadratic field without the generalized Riemann hypothesis is still open. Yokoi conjectured in [9] that there are exactly six real quadratic fields √ of the form Q( m2 + 4) with class number one, which correspond to m =
A. Hoque and S. Kotyada
Arch. Math.
1, 3, 5, 7, 13, 17. In [2], Bir´ o confirmed this conjecture. Recently, Bir´ o and Lapkova [3] extended the result of [2] to a large subclass of Richaud-Degert type real quadratic fields. They proved: Theorem A. For odd positive integers a and m, let d = a2 m2 + 4a. If d is square-free and d > 1253, then h(d) > 1. √ Note that by h(d) we mean the class number of the quadratic field Q( d). As a consequence of Theorem A, one can derive the following: Corollary 1.1. Let d be as in Theorem A. Then 5, 13, 21, 29, 53, 173, 237, 293, 437, 453, 1133, and 1253 are the only values of d such that h(d) = 1. The usual method for proving that class numbers of real quadratic fields K in some parametrized families are greater than 1 is based on continued fractions (cf. [6,7]). This method only works for families for which the continued √ fraction expansion of the canonical generator ω = (dK + dK )/2 of the ring of algebraic integers of K of discriminant dK is known beforehand, say for √ the Q( m2 + 1)’s with m2 + 1 square-free. Another method is based on the computation (in two ways) of special values of the zeta function attached to real quadratic fields. However, this method only works for those fields whose fundamental unit is explicitly known w
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