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We derive midpoint criteria for solving Pell's equation x 2 − Dy 2 = ±1, using the nearest square continued fraction expansion of √ D. The period of the expansion is on average 70% that of the regular continued fraction. We derive similar criteria for the diophantine equation x 2 − xy − (D−1) 4 y 2 = ±1, where D ≡ 1 (mod 4). We also present some numerical… (More)

There are some minor errors in one of the algorithms and two of the tables in a paper by Williams and Buhr. These errors do not affect the major conclusions of the paper. We present corrections to one of the algorithms and two of the tables in [1]. These corrections do not affect the major conclusions of the paper. In the algorithm for computing the NICF of… (More)

1. INTRODUCTION. For primes that can be written as a sum of integer squares, p = a 2 + (2b) 2 , Kaplansky [4] asked whether the binary quadratic form F = x 2 − py 2 always represents a and 4b (that is, are there integer solutions to x 2 − py 2 = a and x 2 − py 2 = 4b). Feit [1] and Mollin [4] proved that F does always represent a and 4b using the theory of… (More)

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