The Somos-5 sequence is defined by and for We relate the arithmetic of the Somos-5 sequence to the elliptic curve . and use properties of Galois representations attached to to prove the density of primes dividing some term in the Somos-5 sequence is equal to .
1. Introduction and statement of results
There are many results in number theory that relate to a determination of the primes dividing some particular sequence. For example, it is well known that if is a prime number, then divides some term of the Fibonacci sequence, defined by , and , for Students in elementary number theory learn that a prime . divides a number of the form if and only if or .
In 1966, Hasse proved in Reference 4 that if is the number of primes so that for some then ,
Note that a prime number divides if and only if has even order in .
A related result is the following. The Lucas numbers are defined by , and for In 1985, Lagarias proved (see .Reference 9 and Reference 10) that the density of primes dividing some Lucas number is Given a prime number . let , be the smallest integer so that A prime . divides for some if and only if is even. In Reference 2, Paul Cubre and the third author prove a conjecture of Bruckman and Anderson on the density of primes for which for an arbitrary positive integer ,.
In the early 1980s, Michael Somos discovered integer-valued non-linear recurrence sequences. The Somos- sequence is defined by and
for Despite the fact that division is involved in the definition of the Somos sequences, the values . are integral for Fomin and Zelevinsky .Reference 3 show that the introduction of parameters into the recurrence results in the being Laurent polynomials in those parameters. Also, Speyer Reference 15 gave a combinatorial interpretation of the Somos sequences in terms of the number of perfect matchings in a family of graphs.
Somos-4 and Somos-5 type sequences are also connected with the arithmetic of elliptic curves (a connection made quite explicit by A. N. W. Hone in Reference 5, and Reference 6). If is the term in the Somos-4 sequence, th and then the denominator of the , of -coordinate is equal to It follows from this that . if and only if reduces to the identity in and so a prime , divides a term in the Somos-4 sequence if and only if has odd order. In Reference 8, Rafe Jones and the third author prove that the density of primes dividing some term of the Somos-4 sequence is The goal of the present paper is to prove an analogous result for the Somos-5 sequence. .
Let denote the number of primes so that divides some term in the Somos-5 sequence. We have the following table of data:
Our main result is the following.
The Somos-5 sequence is related to the coordinates of rational points on the elliptic curve This curve has . and generators are (of infinite order) and (of order We have (see Lemma ).3) that
It follows that a prime divides a term in the Somos-5 sequence if and only if the reduction of modulo is in Another way of stating this is the following: there is a . -isogeny where , and
The kernel of is Letting . we show (see Theorem 4) that a prime of good reduction divides some term in the Somos-5 sequence if and only if the order of in is twice that of in .
A result of Pink (see Proposition 3.2 on page 284 of Reference 11) shows that the valuation of the order of a point -adic can be determined from a suitable Galois representation attached to an elliptic curve. For a positive integer we let , be the field obtained by adjoining to the and coordinates of all points with There is a Galois representation . and we relate the power of dividing the order of in to where , is a Frobenius automorphism at in Using the isogeny . we are able to relate and obtaining a criterion that indicates when , divides some term in the Somos-5 sequence. We then determine the image of for all .
Once the image of is known, the problem of computing the fraction of elements in the image with the desired properties is quite a difficult one. We introduce a new and simple method for computing this fraction and apply it to prove Theorem 1.
If is an elliptic curve given in the form the set , has the structure of an abelian group. Specifically, if let , be the third point of intersection between and the line through and We define . The multiplication by . map on an elliptic curve has degree and so if , is an elliptic curve and then there are , points so that .
If is a finite extension, let denote the ring of algebraic integers in A prime . ramifies in if and some where the , are distinct prime ideals of .
Suppose is Galois, is a prime number that does not ramify in and , For each . there is a unique element , for which
for all This element is called the Artin symbol of . and is denoted If . , and are conjugate in and is a conjugacy class in .
The key tool we will use in the proof of Theorem 1 is the Chebotarev density theorem.
Roughly speaking, each element of arises as equally often.
Let be the set of points of order dividing on Then . is Galois and is isomorphic to a subgroup of Moreover, Proposition V.2.3 of .Reference 13 implies that if is a Frobenius automorphism at some prime above and is the usual mod Galois representation, then and Another useful fact is the following. If . is a number field, is a prime ideal in above , and is not the identity, then does not reduce to the identity in This is a consequence of Proposition VII.3.1 of .Reference 13.
We will construct Galois representations attached to elliptic curves with images in Elements of such a group can be thought of either as pairs . where , is a row vector, and or as , matrices where , and In the former notation, the group operation is given by .
3. Connection between the Somos-5 sequence and
Let be given by and let We have a . -isogeny given by
The elliptic curves and each have conductor The next result classifies the primes of good reduction that divide a term in the Somos-5 sequence. .