## Primes In Divisibility Sequences

Citations: | 8 - 0 self |

### BibTeX

@MISC{Everest_primesin,

author = {Graham Everest and Thomas Ward},

title = {Primes In Divisibility Sequences},

year = {}

}

### OpenURL

### Abstract

We give an overview of two important families of divisibility sequences: the Lehmer{Pierce family (which generalise the Mersenne sequence) and the elliptic divisibility sequences. Recent computational work is described, as well as some of the mathematics behind these sequences.

### Citations

882 |
The Arithmetic of Elliptic Curves
- Silverman
- 1986
(Show Context)
Citation Context ...s Another approach to elliptic divisibility sequences is to start with an elliptic curve (10) y 2 = x 3 + ax + b with coefficients a, b in Z. An excellent reference for this topic is Silverman’s boo=-=k [13]. -=-We must always suppose that the quantity 4a 3 + 27b 2 �= 0. This condition is equivalent to the cubic polynomial x 3 + ax + b having no repeated zeros. If Q = (x, y) is an integer point on the curve... |

112 |
Factorization of certain cyclotomic functions
- Lehmer
(Show Context)
Citation Context ... conjecture. Consider a monic polynomial f(x) = x d + ad−1x d−1 + · · · + a0 with integer coefficients, which factorizes over C as (2) f(x) = (x − α1) . . . (x − αd). Following Pierce [11=-=] and Lehmer [9], we can -=-associate a sequence of integers to f by defining (3) ∆n(f) = d� |α n i − 1| for n ≥ 1. i=1sPRIMES IN SEQUENCES 3 To see that all the terms are integers, let Af denote the companion matrix of... |

107 |
Sequences of numbers generated by addition in formal groups and new primality and factorization tests
- Chudnovsky, Chudnovsky
- 1986
(Show Context)
Citation Context ...ng a similar sort of problem to the one Lehmer faced. Can we find sequences with arbitrarily small growth rate? We are going to discuss this elliptic analogue of Lehmer’s problem later. In the paper=-=s [4]-=- and [5], Chudnovsky and Chudnovsky considered the arithmetic of elliptic divisibility sequences. The following examples are taken from [4]. The first 5 terms are specified, then the first 100 terms a... |

56 |
Memoir on elliptic divisibility sequences
- Ward
- 1948
(Show Context)
Citation Context ...easons, we restrict attention to sequences that have u0 = 1, u1 = 1, u2u3 �= 0 and u2|u4; call these sequences proper. Morgan Ward studied many properties of proper elliptic divisibility sequences i=-=n [14]. Late-=-r on, we will explain in what sense these sequences are ‘elliptic’ – it might seem surprising since there do not seem to be any ellipses on show! The recurrence relation (7) is less straightforw... |

7 |
Computer assisted number theory with applications. Number theory
- Chudnovsky, Chudnovsky
- 1987
(Show Context)
Citation Context ...ilar sort of problem to the one Lehmer faced. Can we find sequences with arbitrarily small growth rate? We are going to discuss this elliptic analogue of Lehmer’s problem later. In the papers [4] an=-=d [5]-=-, Chudnovsky and Chudnovsky considered the arithmetic of elliptic divisibility sequences. The following examples are taken from [4]. The first 5 terms are specified, then the first 100 terms are calcu... |

7 | Primes in Sequences Associated to Polynomials (After
- Einsiedler, Everest, et al.
(Show Context)
Citation Context ...t trivial – that in this case also 1 n log |αn i − 1| → 0. Putting all the three possibilities together and using some transcendence theory to give error estimates, the following can be obtaine=-=d (see [6] or [8-=-]). Theorem 2. There are constants mf ≥ 0, and A = A(f) > 0, such that 1 n log ∆n(f) = mf + O((log n) A /n).s4 GRAHAM EVEREST AND THOMAS WARD The constant mf is given by the formula, d� mf = log... |

7 | The Fibonacci numbers and the Arctic Ocean - Ribenboim - 1995 |

6 | New Fibonacci and Lucas primes
- DUBNER, KELLER
- 1999
(Show Context)
Citation Context ... well-known sequences which satisfy (6). For example the Lucas Sequence satisfies the same recurrence but starts 1, 3, 4, 7 . . . . We call this sequence L with the nth term denoted Ln. The reference =-=[3]-=- gives up-to-date information about primes in these sequences. It seems plausible that a divisibility sequence which satisfies a linear recurrence relation should always contain infinitely many primes... |

3 |
der Poorten, ‘A full characterisation of divisibility sequences
- Bézivin, Pethő, et al.
- 1990
(Show Context)
Citation Context ...even terms of the Fibonacci sequence) contains only one prime, even though it is a divisibility sequence with u1 = 1 and it satisfies a linear recurrence relation. Using the characterization given in =-=[1]-=-, it is possible to explain precisely when a linear divisibility sequence will have generic factorization. What we are saying is that provided this generic factorization is taken into account, one exp... |

3 |
Computational aspects of elliptic divisibility sequences
- Einsiedler, Everest, et al.
(Show Context)
Citation Context ...hether checking for prime appearance was feasible. Thus, the first natural question which occurs is to decide the growth rate for a proper elliptic divisibility sequence. We answered this question in =-=[7].s8 GR-=-AHAM EVEREST AND THOMAS WARD Theorem 4. Suppose un denotes the nth term of a non-trivial proper elliptic divisibility sequence. There are constants κ ≥ 0 and B such that log |un| = κn 2 + O � (l... |

3 | A dynamical property unique to the Lucas sequence
- PURI, WARD
(Show Context)
Citation Context ... It seems a natural question to ask whether the Fibonacci sequence itself can represent the periodic points of a dynamical system and the answer is no. In fact, the following was proved recently (see =-=[10]-=-). Theorem 9. Suppose U = a, b . . . denotes any sequence of positive integers which satisfies (6). Then U is realizable if and only if b = 3a. In other words U is realizable if and only if it is a mu... |

3 | Numerical factors of the arithmetic forms n i=1 (1 a m i - PIERCE - 1917 |

2 |
Numerical factors of the arithmetic forms � n i=1 (1±αm i
- Pierce
- 1917
(Show Context)
Citation Context ...atural analogous conjecture. Consider a monic polynomial f(x) = x d + ad−1x d−1 + · · · + a0 with integer coefficients, which factorizes over C as (2) f(x) = (x − α1) . . . (x − αd). Foll=-=owing Pierce [11] and Lehm-=-er [9], we can associate a sequence of integers to f by defining (3) ∆n(f) = d� |α n i − 1| for n ≥ 1. i=1sPRIMES IN SEQUENCES 3 To see that all the terms are integers, let Af denote the comp... |

1 | der Poorten, `A full characterisation of divisibility sequences - ezivin, o, et al. - 1990 |