Prime Values of Elliptic Divisibility Sequences

If u[n] denotes the n-th term of an EDS then u[n] grows very quickly. In fact

log|u[n]| ~ cn².

Chudnovsky and Chudnovsky considered some EDSs specified by writing down their first five terms and examined these sequences for prime terms. Since u[n] is a divisibility sequence, which grows so quickly, it is sufficient to examine prime occurence of terms u[n] with n prime. The sequences considered by the Chudnovskys are specified below by giving the first 5 terms, as well as the constant c and the occurrence of prime values for prime n up to 100.

Some of the primes in this table are very large. For example, the term u[79] in the third sequence from the end is a prime with 469 decimal digits. It might look as though we should be able to keep computing terms and find larger and larger primes. But if you run the sequences out to n=500 you find no new primes. In a recent paper, there is a heuristic explanation of why these sequences should stop producing primes beyond a certain point. This also explains why there should be a uniform bound on the number of primes. A proof of finiteness under a hypothesis on 2-torsion is obtainable here.

Other Models

Suppose we ask the same question for an elliptic curve in homogeneous form,

x³ + y³ = c,

for a non-zero rational c. Suppose P is a non-torsion rational point. Write x(nP)=A_n/B_n for integral A_n and B_n. The paper just referred to gives a proof that only finitely many terms B_n are prime.

Zsigmondy's Theorem

Primes From Rational Points

It does look, in some cases, as though rational points on elliptic curves can produce primes in abundance if the Mordell-Weil rank of the curve is greater than 1. Let E denote such a curve, with two independent rational points P and Q. Let M and N denote integers and consider the bi-sequence MP+NQ, the denominator of the x-coordinate is always the square of an integer, let s(M,N) denote the square root of the denominator. We appear to have more joy looking for prime values of this bi-sequence. Simple heuristics (see below) suggest that the number of prime values of the sequence s(M,N) with |M| and |N| bounded by X is approximately

The constant c should depend upon the elliptic regulator. The paper just referred to gives examples where only finitely many primes occur. For curves in homogeneous form, only finitely many primes appear with no extra hypotheses on the torsion. Nailing precise criteria for finiteness is an open problem.

Cremona has a list of curves and points for conductors up to 6000. From these, the first few with rank 2 can be siphoned off to a separate list . Using this list, Peter Rogers is now collecting some data on this problem. Whether the heuristic argument is accurate or not, this method certainly produces some large primes.

1. e = [0,1,0,-25,39], P = [-5,8], Q = [-1,8]: s(118,31) is a prime with 2705 decimal digits. This curve appears as number 26 in the list .
2. e = [0,-1,1,-9,9], P = [1,0], Q = [7,15]: s(111,47) is a prime with 2541 decimal digits. This curve is number 34 on the list.

There is an interesting history of heuristics associated to prime occurrence in integer sequences. Chris Caldwell gives a brief account of heuristics for prime occurrence in the Mersenne Sequence in his interesting Prime Pages. The basic idea is to reckon that the Prime Number Theorem gives the probability 1/logN that the integer N is prime. So if you have an increasing sequence a[n] of positive integers then you would reckon that the number of terms of a[n] with n < X which are prime should be about

summed over n < X. Hardy and Wright pointed out that this kind of argument suggests there should only be finitely many Fermat Primes. In general, this basic argument needs some refinement to make it work. The numbers a[n] could all be even for example in which case the argument would be nonsense! In specific cases, such as Mersenne, refinements can be made to work which fit the data. In that case, Merten's Theorem is used to predict approximately

Mersenne Primes 2ⁿ-1 with n < X, where the constant v is given explicitly by the formula

where g is the Euler-Mascheroni constant. The paper gives a similar refinement for EDSs, using a combination of Merten's Theorem and Hasse's Theorem.

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