Only the leading seven terms of a continued fraction are needed to perform on-line arithmetic, provided the continued fractions are of the correct form. This forms the basis of a proof that there is an effective representation of the computable reals as continued fractions: we also demonstrate that the basic arithmetic operations are computable using this representation.

The continued fraction expansion of a real number is a fundamental and revealing expansion through its connection with Euclidean algorithm and with ``best'' rational approximations (see [HW]). At the same time, it is very poorly understood for some interesting numbers. We know that it is essentially unique and finite (i.e., terminating) exactly for rational numbers and periodic exactly for quad...

The purpose of this article is to provide criteria for the solvability of the Diophantine equation a2X2− bY 2 = c in terms of the simple continued fraction expansion of √ a2b, and to explore criteria for the solvability of AX2 − BY 2 = C for given A, B, C ∈ N in the general case. This continues work in [9]–[11].

If the equation of the title has an integer solution with k ≥ 2, then m > 109.3·10 6 . This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark m > 1010 7 . Here we achieve m > 1010 9 by showing that 2k/(2m−3) is a convergent of log 2 and making an extensive continued fraction digits calculation of (log 2)/N , wit...