A. J. van der Poorten

Folded continued fractions

Abstract We exhibit uncountably many binary decimals together with their explicit continued fraction expansions. These expansions require only the partial quotients 1 or 2. The pattern of valleys and ridges in a sheet of paper repeatedly folded in half plays a critical role in our construction.

Zeros of recurrence sequences

We give an upper bound for the number of zeros of recurrence sequences defined over an algebraic number field in terms of their order, the degree of their field of definition and the number of prime ideal divisors of the characteristic roots of the sequence.

Arithmetic and analytic properties of paper folding sequences

The mechanical procedure of paper folding generates an uncountable family of infinite sequences of fold patterns. We obtain the associated Fourier series and show that the sequences are almost periodic and hence deterministic. Further, we show that paper folding numbers defined by the sequences are all transcendental.

A specialised continued fraction

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 arguments, let alone examples, demonstrate that it is noteworthy if those partial quotients happen to have rational integer coefficients only. In that special case one may replace the variable X by an integer ≥ 2 ; that is: one may ‘specialise’ and thereby proceed to obtain the regular continued fraction expansion of values of the series. And that is significant because, generally, it is difficult to obtain the explicit continued fraction expansion of a number presented in different shape. Our example leads to a series with a specialisable continued fraction expansion and, a little surprisingly, our arguments suggest that the phenomenon of specialisability for series of the kind appearing here may be reserved to just the special subclass of series we happen to have stumbled upon.

SOME PROBLEMS CONCERNING RECURRENCE SEQUENCES

There are questions about recurrence sequences that seem to crop up again and again. Plainly, though their answers are well known they are not known well. We endeavour to explain these answers in context so that they may become more widely known. The sequence 0,1,-1,2,-2,..., in which each integer occurs exactly once, is a recurrence sequence; that is, it satisfies a linear, homogeneous recurrence relation with constant coefficients, namely,

Elliptic curves of conductor 11

We determine all elliptic curves defined over Q of conductor 11. Firstly, we reduce the problem to one of solving a diophantine equation, namely a certain ThueMahler equation. Then we apply recent sharp inequalities for linear forms in the logarithms of algebraic numbers to bound solutions of that equation. Finally, some straightforward computations yield all solutions of the diophantine equation. Our results are in accordance with the conjecture of Taniyama-Weil for conductor 11. Taniyama and Weil have asked whether all elliptic curves defined over Q of a given conductor N are parametrized by modular functions for the subgroup ro(N) of the modular group. The assertion that this question has a positive answer has become known as the Taniyama-Weil conjecture. While the general question seems shrouded in mystery and quite inaccessible at present, one can at least try to verify the conjecture for small numerical values of N. A considerable amount of work has already been done in this direction (cf. [4], [5], [19] -[24], [29]). However, the first nontrivial case of the conjecture, namely N = 11, has not previously been settled. The aim of this note is to determine all elliptic curves of conductor 11 defined over Q and so to verify the conjecture of Taniyama-Weil for N = 11. It is well known that the problem of finding all elliptic curves defined over Q of a given conductor N can be reduced to finding S-integral points on certain associated curves of genus 1; here S is the set of primes dividing N. For certain values of N, these diophantine equations can easily be solved by congruence techniques. However, this elementary approach does not work for N = 11, and we are forced to solve these equations by using some recent sharp inequalities for linear forms in the logarithms of algebraic numbers. The body of this paper is, thus, given over to solving a diophantine equation by Bakers method. Whilst our computations are of course specific to the particular equation we solve, our methods are quite general. As regards the elliptic curves, we employ the usual notation and terminology. For background and more detailed explanation we refer the reader to the surveys of Swinnerton-Dyer and Birch [31] and of Gelbart [12]; see also Mazur and SwinnertonDyer [18]. 1. An elliptic curve E over a field K has a nonsingular plane cubic model (1) y2 + a1xy + a3y = X3 + a2x2 + a4x + a6 Received July 29, 1979; revised September 19, 1979. 1980 Mathematics Subject Classification. Primary 1OF10, 1OD12, 1OB10, 1OB16, 14K07, 12A30. i 1980 American Mathematical Society 0025-5718/80/0000-01 23/

Fractions of the period of the continued fraction expansion of quadratic integers

04.00 991 This content downloaded from 157.55.39.215 on Wed, 31 Aug 2016 04:17:10 UTC All use subject to http://about.jstor.org/terms 992 M. K. AGRAWAL, J. H. COATES, D. C. HUNT AND A. J. VAN DER POORTEN with the ai in K. If the characteristic char K of K is not 2, we can replace 4(2y +a1x + a3) byy and 4x byx to obtain (2) y2 = x3 + b2x2 + 8b4x + 16b6

On strong pseudoprimes in arithmetic progressions

The elementary nature and simplicity of the theory of continued fractions is mostly well disguised in the literature. This makes one reluctant to quote sources when making a remark on the subject and seems to necessitate redeveloping the theory ab initio . That had best be done succinctly. That is done here and allows the retrieval of some amusing results on pattern in the period of the continued fraction expansion of quadratic integers.

On a sequence of prime numbers

A composite integer N is said to be a strong pseudoprime for the base C if with N – 1 = 2 s d , (2, d ) = 1 either C d = 1, or C 2r ≡ 1 (mod N ) some r , 0 ≤ r s . It is shown that every arithmetic progression ax+b ( x = 0,1, …) where a, b are relatively prime integers contains an infinite number of odd strong pseudoprimes for each base C ≤ 2. 1980 Mathematics subject classification (Amer. Math. Soc.) : 10 A 15.

Corrigenda and addition to Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000

Euclids scheme for proving the infinitude of the primes generates, amongst others, the following sequence defined by p 1 = 2 and p n+1 is the highest prime factor of p 1 p 2 … p n +1 .