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We detail the continued fraction expansion of the square root of the general monic quartic polynomial. We note that each line of the expansion corresponds to addition of the divisor at infinity, and interpret the data yielded by the general expansion. The paper includes a detailed ’reminder exposition’ on continued fractions of quadratic irrationals in… (More)

Assuming a weak version of a conjecture of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m > 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m.

We display a number with a surprising continued fraction expansion and show that we may explain that expansion as a specialisation of the continued fraction expansion of a formal series: A series ∑ chX −h has a continued fraction expansion with partial quotients polynomials in X of positive degree (other, perhaps than the 0-th partial quotient). Simple… (More)

w x Ž . In 1948, Paul Erdos E1 proved the irrationality of h 1 . Recently, ̋ 2 w x Peter Borwein used Pade approximation techniques B1 and some coḿ w x Ž . plex analysis methods B2 to prove the irrationality of both h 1 and q Ž . Ln 2 . Here we present a proof in the spirit of Apery’s magnificent proof ́ q Ž . w x of the irrationality of z 3 A , which was… (More)

We provide evaluations of several recently studied higher and multiple Mahler measures using log-sine integrals. This is complemented with an analysis of generating functions and identities for log-sine integrals which allows the evaluations to be expressed in terms of zeta values or more general polylogarithmic terms. The machinery developed is then… (More)

In his ‘Memoir on Elliptic Divisibility Sequences’, Morgan Ward’s definition of the said sequences has the remarkable feature that it does not become at all clear until deep into the paper that there exist nontrivial such sequences. Even then, Ward’s proof of coherence of his definition relies on displaying a sequence of values of quotients of Weierstraß σ… (More)

with θ irrational and taken to satisfy 0 < θ < 1; plainly this may be assumed without loss of generality. Evidently (fn) is a sequence of zeros and ones. Denote by w0 and w1 words on the alphabet {0, 1} ; that is, finite strings in the letters 0 and 1. Then the sequence (fn) is said to be invariant under the substitution W given by W : 0 −→ w0, 1 −→ w1, if… (More)

- AARON EKSTROM, Alf van der Poorten
- 2011

In 1987, Gordon gave an integer primality condition similar to the familiar test based on Fermat’s little theorem, but based instead on the arithmetic of elliptic curves with complex multiplication. We prove the existence of infinitely many composite numbers simultaneously passing all elliptic curve primality tests assuming a weak form of a standard… (More)

Introduction. The purpose of these notes is to support the papers included in this volume by providing self-contained summaries of related mathematics emphasising those aspects actually used or required. I have selected two topics that are easily described from first principles yet for which it is peculiarly difficult to find congenial introductions that… (More)

We describe a variant of Fermat’s factoring algorithm which is competitive with SQUFOF in practice but has heuristic run time complexity O(n1/3) as a general factoring algorithm. We also describe a sparse class of integers for which the algorithm is particularly effective. We provide speed comparisons between an optimised implementation of the algorithm… (More)