The density of primes dividing a term in the Somos-5 sequence

By Bryant Davis, Rebecca Kotsonis, Jeremy Rouse

Abstract

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 th term in the Somos-4 sequence, and , then the denominator of the -coordinate of 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.

Theorem 1.

We have

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 -adic valuation of the order of a point 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.

2. Background

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.

Theorem 2 (Reference 7, page 143).

If is a conjugacy class, then

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

Lemma 3.

Define and on . For all , we have the following relationship between the Somos-5 sequence and :

Proof.

We will prove this by strong induction. A straightforward calculation shows that the base cases and are true. For simplicity’s sake, we will denote , , , , , , , and . Our inductive hypothesis is that

We will now compute .

To find the and coordinates of , we add to . If is the slope and is the -intercept, the line between and is with and . Substituting this into the equation for , we find the -coordinate of to be . A straightforward but lengthy inductive calculation shows that if

then for all . Also, holds (by Proposition 2.8 in Hone’s paper Reference 6). Since , we know that . Therefore, we know that .

Denote the -coordinate of as . We compute that . Using that , we find that . Therefore, it is evident that

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.

Theorem 4.

If is a prime of good reduction that divides a term in the Somos-5 sequence, the order of in is twice the order of in . Otherwise, their orders are the same.

Proof.

If divides a term in our sequence, say , we know from our previous lemma that the denominators are divisible by . Therefore, modulo , . The point has order 2, so adding to both sides we know that . Therefore, we can deduce that . We have (see Section 3.4 of Reference 14). Therefore, if is restricted to the subgroup generated by , we have . Since