On the divisibility of class numbers of quadratic fields and the solvability of diophantine equations

Abstract

In this paper we provide criteria for the insolvability of the Diophantine equationThis result is then used to determine the class number of the quadratic field ℚ √− .We also determine some criteria for the divisibility of the class number of the quadratic field ℚ √− and this result is then used to discuss the solvability of the Diophantine equation

Complete solutions of certain Lebesgue--Ramanujan--Nagell type equations

“…There are many interesting results concerning the solutions of (1.4) for λ = 1. For λ = 1, this equation has been extensively studied by many authors (see [4], [11] and [16]- [18]), and thus there are many interesting results concerning its solutions. The equations of the form (1.1) are well connected with the investigation of the class number of imaginary quadratic field Q( √ −D).…”

“…While (λ, C) = (1, p a 1 1 ) the equation (1.1) is completely settled by the combined work of several authors (cf. [2,4,5,11,17,20]). For (λ, C) = (2, p a 1 1 ) with even a 1 , (1.1) is studied in [19,28].…”

Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations

“…The first result in this case is due to V. A. Lebesgue [22], who proved that (1.3) has no solutions in positive integers x, y and n when D = 1. For D > 1 and for λ = 1, the same equation has been extensively studied by several authors and in particular, by J. H. E. Cohn [15,14], A. Hoque and H. K. Saikia [18], and M. Le [19,20]. We also refer to [14] for a complete survey on (1.3) when λ = 1.…”

On the solutions of a Lebesgue–Nagell type equation

“…The first result in this case is due to V. A. Lebesgue [18], who proved that (1.1) has no solutions in positive integers x, y and n when D = 1. For D > 1 and for λ = 1, (1.1) has been extensively studied by several authors and in particular, by J. H. E. Cohn [15,16], A. Hoque and H. K. Saikia [17], and M. Le [9,10]. We also refer to [16] for a complete survey on (1.1) when λ = 1.…”