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A Novel Deterministic Mersenne Prime Numbers Test: Aouessare-El Haddouchi-Essaaidi Primality Test

2014
*
International Journal of Computer Applications
*

*The*best method presently known for

*Mersenne*numbers primality

*testing*is

*Lucas*-

*Lehmer*primality

*test*. ...

*The*largest

*prime*number discovered so far, which is a

*Mersenne*number, has 17,425,170 digits. However,

*the*algorithmic complexity of

*Mersenne*

*primes*

*test*is computationally very expensive. ...

*The*best method so far known

*and*widely used for

*testing*

*the*primality of

*Mersenne*numbers is

*Lucas*-

*Lehmer*primality

*test*. ...

##
###
A New Mersenne Prime

1991
*
Mathematics of Computation
*

*The*number 2 -1 is a

*Mersenne*

*prime*. There are exactly two

*Mersenne*exponents between 100000

*and*139268,

*and*there are no

*Mersenne*... Acknowledgments

*The*authors thank

*the*Houston Area Research Center for donation of computer time. ... We are also grateful to

*the*anonymous referee whose suggestions made this paper far better than it would have been otherwise. Bibliography ...

##
###
A new Mersenne prime

1991
*
Mathematics of Computation
*

*The*number 2 -1 is a

*Mersenne*

*prime*. There are exactly two

*Mersenne*exponents between 100000

*and*139268,

*and*there are no

*Mersenne*... Acknowledgments

*The*authors thank

*the*Houston Area Research Center for donation of computer time. ... We are also grateful to

*the*anonymous referee whose suggestions made this paper far better than it would have been otherwise. Bibliography ...

##
###
The 25th and 26th Mersenne Primes

1980
*
Mathematics of Computation
*

*The*25th

*and*26th

*Mersenne*

*primes*are 221701 -1

*and*223209 -1, respectively. Their primality was determined with an implementation of

*the*

*Lucas*-

*Lehmer*

*test*on a CDC Cyber 174 computer. ...

*The*25th

*and*26th even perfect numbers ...

*Lehmer*...

##
###
The 25th and 26th Mersenne primes

1980
*
Mathematics of Computation
*

*The*25th

*and*26th

*Mersenne*

*primes*are 221701 -1

*and*223209 -1, respectively. Their primality was determined with an implementation of

*the*

*Lucas*-

*Lehmer*

*test*on a CDC Cyber 174 computer. ...

*The*25th

*and*26th even perfect numbers ...

*Lehmer*...

##
###
Mersenne and Fermat Numbers

1954
*
Proceedings of the American Mathematical Society
*

Such a

doi:10.2307/2031878
fatcat:mynlevnwczfuhdjkb7wobdqosu
*test*was first applied by*Lucas*in 1876 to show that 2m -1 is*prime*. ...*The*first twelve of*the**Mersenne**primes*have been known since 1914;*the*twelfth, 2127 -1, was indeed found by*Lucas*as early as 1876,*and*for*the*next seventy-five years was*the*largest known*prime*. ...##
###
Mersenne and Fermat numbers

1954
*
Proceedings of the American Mathematical Society
*

Such a

doi:10.1090/s0002-9939-1954-0064787-4
fatcat:kxuwnzc3t5h77ccbg75eitx3ve
*test*was first applied by*Lucas*in 1876 to show that 2m -1 is*prime*. ...*The*first twelve of*the**Mersenne**primes*have been known since 1914;*the*twelfth, 2127 -1, was indeed found by*Lucas*as early as 1876,*and*for*the*next seventy-five years was*the*largest known*prime*. ...##
###
New Mersenne primes

1962
*
Mathematics of Computation
*

If p is

doi:10.1090/s0025-5718-1962-0146162-x
fatcat:6rqrmmytrfcytcc4747hstyqpm
*prime*, Mp = 2V -1 is called a*Mersenne*number.*The**primes*Af4263*and*Mmz were discovered by coding*the**Lucas*-*Lehmer**test*for*the*IBM 7090. ... This*test*is described by*the*following theorem (see*Lehmer*[1,). Theorem. If Si = á*and*Sn+i = Sn2 -2 then Mv is*prime*if*and*only if Sp-i = 0 (mod Mp). ...*The**Lucas*-*Lehmer**test*can also be used with Si = 10.*The*various penultimate residues of*the*known*Mersenne**primes*were computed*and**the*results appear in Table 1 (see Robinson [3] ). ...##
###
New Mersenne Primes

1962
*
Mathematics of Computation
*

If p is

doi:10.2307/2003068
fatcat:3qtx4oojonhhtizeqjob72slfy
*prime*, Mp = 2V -1 is called a*Mersenne*number.*The**primes*Af4263*and*Mmz were discovered by coding*the**Lucas*-*Lehmer**test*for*the*IBM 7090. ... This*test*is described by*the*following theorem (see*Lehmer*[1,). Theorem. If Si = á*and*Sn+i = Sn2 -2 then Mv is*prime*if*and*only if Sp-i = 0 (mod Mp). ...*The**Lucas*-*Lehmer**test*can also be used with Si = 10.*The*various penultimate residues of*the*known*Mersenne**primes*were computed*and**the*results appear in Table 1 (see Robinson [3] ). ...##
###
Taxonomy and Practical Evaluation of Primality Testing Algorithms
[article]

2020
*
arXiv
*
pre-print

In this paper, an intensive survey is thoroughly conducted among

arXiv:2006.08444v1
fatcat:ndiycg36arh3rk6fau72nck5iu
*the*several primality*testing*algorithms showing*the*pros*and*cons,*the*time complexity,*and*a brief summary of each algorithm. ... Primality*testing*algorithms are used to determine whether a particular number is*prime*or composite. ...*Lucas*-*Lehmer*Theorem states: let M p = 2 p − 1 be a*Mersenne*number to be*tested*where p an odd*prime*. ...##
###
Fast Mersenne prime testing on the GPU

2011
*
Proceedings of the Fourth Workshop on General Purpose Processing on Graphics Processing Units - GPGPU-4
*

*The*

*Lucas*-

*Lehmer*

*test*for

*Mersenne*primality can be efficiently parallelized for GPU-based computation. ... Results show up to 7× speedups over benchmark averages for optimized sequential code

*and*factor-oftwo speedups over CUDALucas, another GPU-based

*Lucas*-

*Lehmer*tester developed independently

*and*with a different ... Greg Childers

*the*many others on mersenneforum.org who have provided information on current software

*and*offers of help with

*the*

*testing*

*and*distribution of gpuLucas. ...

##
###
Primality tests for 2^kn-1 using elliptic curves
[article]

2009
*
arXiv
*
pre-print

Gross already proved

arXiv:0912.5279v1
fatcat:45de4mptpbdeveyf4x67ssuzr4
*the*same result about a primality*test*for*Mersenne**primes*using elliptic curve. ... Essentially,*the*new primality*tests*are*the*elliptic curve version of*the**Lucas*-*Lehmer*-Riesel primality*test*. Note:An anonymous referee suggested that Benedict H. ... When n is relatively small as in*the**Lucas*-*Lehmer*-Riesel*test*,*the*primality*test*can be regarded as an analogue of*the**Lucas*-*Lehmer*-Riesel*test*. ...##
###
Biquadratic reciprocity and a Lucasian primality test

2003
*
Mathematics of Computation
*

Then, for a suitable seed s 0 ,

doi:10.1090/s0025-5718-03-01575-8
fatcat:i6xesklpvfbx7mwsjcs7zxcl5e
*the*number In general s 0 depends both on h*and*on n. ... In particular, when h = 4 m − 1, m odd, we have a*test*with a single seed depending only on h, in contrast with*the*unmodified*test*, which, as proved by W. ...*The*terminology comes from*the**Lucas*-*Lehmer**test*for*Mersenne**primes*(see [4] for historical details): (1) M is*prime*. (2) (α/α) (M+1)/2 ≡ −1 mod M . (3) s n−2 ≡ 0 mod M , where s k is*the*Lucasian sequence ...##
###
Chebyshev polynomials and higher order Lucas Lehmer algorithm
[article]

2021
*
arXiv
*
pre-print

We extend

arXiv:2010.02677v2
fatcat:j3cqze2r6nb2znu7ldkjoru5ra
*the*necessity part of*Lucas**Lehmer*iteration for*testing**Mersenne**prime*to all base*and*uniformly for both generalized*Mersenne**and*Wagstaff numbers(*the*later correspond to negative base). ... This results from a Chebyshev polynomial primality*test*based essentially on*the**Lucas*pair $(\omega_a,\overline{\omega}_a)$, $\omega_a=a+\sqrt{a^2-1}$, where $a \neq 0 \pm 1$. ...*The*q-nary*Lucas**Lehmer*is essentially known in many posting by Pedja Terzić [5, 6]*and*these can all be derived from our main Lemma 1.6. ...##
###
An elliptic curve test for Mersenne primes

2005
*
Journal of Number Theory
*

Let 3 be a

doi:10.1016/j.jnt.2003.11.011
fatcat:tdqa2so5yjfibkppmlpaifgxhq
*prime*,*and*let p = 2 − 1 be*the*corresponding*Mersenne*number.*The**Lucas*-*Lehmer**test*for*the*primality of p goes as follows. Define*the*sequence of integers x k by*the*recursion ... Acknowledgments It is a pleasure to thank Hendrik Lenstra, who first introduced elliptic curves into*the*field of primality*testing*, for his suggestions. ...*Lucas*-*Lehmer*If 3 is a*prime*,*and*p = 2 − 1 is*the*corresponding*Mersenne*number, then p ≡ 7 (mod 24). (1.1) We will exploit this congruence throughout*the*paper. ...
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