An elliptic curve test for Mersenne primes
type: Item
instance of: scholarly article
title: An elliptic curve test for Mersenne primes
zbMATH Open document ID: 1074.11065
published in: Journal of Number Theory
publication date: 2005-03-08T00:00:00Z
review text: Let \(p=2^l-1\), \(l\) an odd prime, be the \(l\)th Mersenne number. The author first derives the Lucas-Lehmer test following ideas of M. Rosen, by showing that \(p\) is prime if and only if the image of \(\epsilon= 2+\sqrt{3}\) in the group \(({\mathbb Z}+{\mathbb{Z}}\sqrt{3}/p)^*\) has order \(2^l\). The Lucas-Lehmer test follows by looking at the traces of \(\varepsilon^{2^k}\) mod \(p\), \(0 \leq k \leq l-2\). Motivated by this, the author next shows that if \(E\) is the elliptic curve \(y^2=x(x^2-12)\) and \(P\) is the point \((2,4)\) on \(E\), (which has infinite order in \(E({\mathbb Q})\)), then the Mersenne number \(p\) is prime if and only if \(P\) has order \(2^l\) in \(E(p)\), where \(E(p)\) denotes the \({\mathbb Z}/p\)-valued points of \(E\). Taking \(x\)-coordinates of the points \(2^kP\) gives a recursively defined sequence \(\{x_k\}\) for which the following holds: if \(p\) is prime then the rational numbers \(x_k(x_k^2-1)\) are \(p\)-adic units and \(x_{l-1}\equiv 0 \) (mod \(p\)). Conversely, if the \(x_k(x_k^2-12)\) are relatively prime to \(p\) for \(0 \leq k \leq l-2\) and \(\gcd(x_{l-1},p)>1\) then \(p\) is prime. The author asks whether these two very similar-looking tests, both involving algebraic groups, are related, and whether other tests can be developed using different algebraic groups -- a challenge to the knowledgeable and astute members of the elliptic curve community. The work of P. Berrizbeitia and collaborators (including this reviewer) in fact implicitly uses other algebraic tori coming from number fields. See, e.g. [\textit{P. Berrizbeitia, B. Iskra}, ``Deterministic primality test for numbers of the form \(A^2. 3^n+1\), \(n \geq 3\) odd'', Proc. Am. Math. Soc. 130, No. 2, 363--365 (2002; Zbl 1006.11077), \textit{P. Berrizbeitia, T. G. Berry, J. Tena-Ayuso}, ``A generalization of Proth's theorem'', Acta Arith. 110, No. 2, 107--115 (2003; Zbl 1028.11001)]. Finally, the reviewer notes that the Lucas-Lehmer test as stated in the pr…
Mathematics Subject Classification ID: 11Y11
zbMATH DE Number: 2142239
zbMATH Keywords: Mersenne prime
zbMATH Keywords: elliptic curve
author: Benedict H. Gross
reviewed by: T. G. Berry
describes a project that uses: ecdata
MaRDI profile type: Publication
full work available at URL: j.jnt.2003.11.011
OpenAlex ID: W1996533598
cites work: Q4003857
cites work: A Hyperelliptic Smoothness Test, II
cites work: A Proof of the Lucas-Lehmer Test
cites work: On the distribution of integer points on curves of genus zero
cites work: Q4317891
DOI: 10.1016/J.JNT.2003.11.011
description: scientific article; zbMATH DE number 2142239