Primes generated by elliptic curves

Everest, Graham; Miller, Victor S.; Stephens, N. M.

doi:10.1090/s0002-9939-03-07311-8

Cited by 34 publications

(56 citation statements)

References 11 publications

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“…However, a heuristic argument (discussed later) using the prime number theorem was adapted in [7], and it suggested that a nonlinear Somos sequence should contain only finitely many prime terms. See [9] for proofs in some special cases.…”

Section: Is Through the (Binary) Linear Recurrence Relationmentioning

confidence: 99%

Primes Generated by Recurrence Sequences

Everest

Stevens

Tamsett³

et al. 2007

The American Mathematical Monthly

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MERSENNE NUMBERS AND PRIMITIVE PRIME DIVISORS.A notorious problem from elementary number theory is the "Mersenne Prime Conjecture." This asserts that the Mersenne sequence M = (M n ) defined by M n = 2 n − 1 (n = 1, 2, . . . ) contains infinitely many prime terms, which are known as Mersenne primes.The Mersenne prime conjecture is related to a classical problem in number theory concerning perfect numbers. A whole number is said to be perfect if, like 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14, it is equal to the sum of all its proper divisors. Euclid pointed out that 2 k−1 (2 k − 1) is perfect whenever 2 k − 1 is prime. A much less obvious result, due to Euler, is a partial converse: if n is an even perfect number, then it must have the form 2 k−1 (2 k − 1) for some k with the property that 2 k − 1 is a prime. Whether there are any odd perfect numbers remains an open question. Thus finding Mersenne primes amounts to finding (even) perfect numbers.The sequence M certainly produces some primes initially, for example,However, the appearance of Mersenne primes quickly thins out: only forty-three are known, the largest of which, M 30,402,457 , has over nine million decimal digits. This was discovered by a team at Central Missouri State University as part of the GIMPS project [23], which harnesses idle time on thousands of computers all over the world to run a distributed version of the Lucas-Lehmer test. A paltry forty-three primes might seem rather a small return for such a huge effort. Anybody looking for gold or gems with the same level of success would surely abandon the search. It seems fair to ask why we should expect there to be infinitely many Mersenne primes. In the absence of a rigorous proof, our expectations may be informed by heuristic arguments. In section 3 we discuss heuristic arguments for this and other more or less tractable problems in number theory.Primitive prime divisors. In 1892, Zsigmondy [24] discovered a beautiful argument that shows that the sequence M does yield infinitely many prime numbers-but in a less restrictive sense. Given any integer sequence S = 1

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Section: Is Through the (Binary) Linear Recurrence Relationmentioning

confidence: 99%

Primes Generated by Recurrence Sequences

Everest

Stevens

Tamsett³

et al. 2007

The American Mathematical Monthly

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“…Somos 4 sequences, and EDS in particular, are of considerable interest to number theorists due to the way that primitive prime divisors appear [6,7,9,10,11,12,35]. In that and in other respects, they have a lot in common with linear recurrence sequences (see [8, chapters 10 and 11]).…”

Section: Introductionmentioning

confidence: 99%

Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences

Hone

Swart

2008

Math. Proc. Camb. Phil. Soc.

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Somos 4 sequences are a family of sequences defined by a fourth-order quadratic recurrence relation with constant coefficients. For particular choices of the coefficients and the four initial data, such recurrences can yield sequences of integers. Fomin and Zelevinsky have used the theory of cluster algebras to prove that these recurrences also provide one of the simplest examples of the Laurent phenomenon: all the terms of a Somos 4 sequence are Laurent polynomials in the initial data. The integrality of certain Somos 4 sequences has previously been understood in terms of the Laurent phenomenon. However, each of the authors of this paper has independently established the precise correspondence between Somos 4 sequences and sequences of points on elliptic curves. Here we show that these sequences satisfy a stronger condition than the Laurent property, and hence establish a broad set of sufficient conditions for integrality. As a by-product, non-periodic sequences provide infinitely many solutions of an associated quartic Diophantine equation in four variables. The analogous results for Somos 5 sequences are also presented, as well as various examples, including parameter families of Somos 4 integer sequences.

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“…Building on the heuristic argument above, Einsiedler et al [10] conjectured that an EDS has only finitely many prime terms, and this conjecture was later expanded upon by Everest et al [13]. For some EDSs, finiteness follows from a type of generic factorization not unlike (2.1) (see, for example, [13,14,16,26] and Section 6), but the general case appears difficult. The study of primitive divisors in integral Lucas sequences goes back to the 19th-century work of Bang [2] and Zsigmondy [57], who showed that a n − b n has a primitive divisor for all n > 6.…”

Section: History and Motivationmentioning

confidence: 99%

Algebraic Divisibility Sequences Over Function Fields

Ingram

Mahé

Silverman

et al. 2012

J. Aust. Math. Soc.

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In this note we study the existence of primes and of primitive divisors in function field analogues of classical divisibility sequences. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.2010 Mathematics subject classification: primary 11B39; secondary 11G05. Keywords and phrases: lucas sequence, elliptic divisibility sequence, primitive divisor, function field over number field.

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Primes generated by elliptic curves

Cited by 34 publications

References 11 publications

Primes Generated by Recurrence Sequences

Primes Generated by Recurrence Sequences

Integrality and the Laurent phenomenon for Somos 4 and Somos 5 sequences

Algebraic Divisibility Sequences Over Function Fields

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