On a Generalized Kaplansky Conjecture
نویسنده
چکیده
A conjecture was related to this author in correspondence, some years ago, with Irving Kaplansky, which according to Professor Kaplansky, was inspired by the proof of [4, Theorem 6.5.9, p. 348]. It asserts that if p is a prime with representation p = a2 + (2b)2, then the equation x2 − py2 = a is solvable in integers x, y. In [5], we proved this conjecture along with several others by him. Subsequently, Walsh in [6], gave a slight extension of the above proof: if n ≡ 1 (mod 4) is a nonsquare integer with representation n = a2 + (2b)2 for integers a and b, and if X2 − nY 2 = −1 has solutions in integers X,Y , then n has a factorization n = rs such that the equation ru2 − sv2 = a is solvable in integers u, v. It is the purpose of this work to generalize the latter to a much wider range of cases as given in Theorem 1.1 below. We illustrate with several examples to show the wide applicability of the result. Mathematics Subject Classification: 11A51, 11D09, 11R11
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