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

Abdelilah Aouessare, Abdeslam El haddouchi, Mohamed Essaaidi
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.  ... 
doi:10.5120/17505-8053 fatcat:2nsyxfvdl5b6fnorcodhmpqh5a

A New Mersenne Prime

W. N. Colquitt, L. Welsh
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  ... 
doi:10.2307/2008415 fatcat:aek7iqhh6vfz5fy2s4ntd7nsr4

A new Mersenne prime

W. N. Colquitt, L. Welsh
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  ... 
doi:10.1090/s0025-5718-1991-1068823-9 fatcat:ten3yzmharhrhk72qylyp4rjdu

The 25th and 26th Mersenne Primes

Curt Noll, Laura Nickel
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  ... 
doi:10.2307/2006405 fatcat:j5lmlmdiljbyrpbfxmf3xz22w4

The 25th and 26th Mersenne primes

Curt Noll, Laura Nickel
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  ... 
doi:10.1090/s0025-5718-1980-0583517-4 fatcat:drqengf4pnekrmnwldnzg3agd4

Mersenne and Fermat Numbers

Raphael M. Robinson
1954 Proceedings of the American Mathematical Society  
Such a 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.  ... 
doi:10.2307/2031878 fatcat:mynlevnwczfuhdjkb7wobdqosu

Mersenne and Fermat numbers

Raphael M. Robinson
1954 Proceedings of the American Mathematical Society  
Such a 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.  ... 
doi:10.1090/s0002-9939-1954-0064787-4 fatcat:kxuwnzc3t5h77ccbg75eitx3ve

New Mersenne primes

Alexander Hurwitz
1962 Mathematics of Computation  
If p is 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] ).  ... 
doi:10.1090/s0025-5718-1962-0146162-x fatcat:6rqrmmytrfcytcc4747hstyqpm

New Mersenne Primes

Alexander Hurwitz
1962 Mathematics of Computation  
If p is 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] ).  ... 
doi:10.2307/2003068 fatcat:3qtx4oojonhhtizeqjob72slfy

Taxonomy and Practical Evaluation of Primality Testing Algorithms [article]

Anas AbuDaqa, Amjad Abu-Hassan, Muhammad Imam
2020 arXiv   pre-print
In this paper, an intensive survey is thoroughly conducted among 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.  ... 
arXiv:2006.08444v1 fatcat:ndiycg36arh3rk6fau72nck5iu

Fast Mersenne prime testing on the GPU

Andrew Thall
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.  ... 
doi:10.1145/1964179.1964188 dblp:conf/asplos/Thall11 fatcat:h4skd5lnvvbtpjwb3nkknbbpne

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

Yu Tsumura
2009 arXiv   pre-print
Gross already proved 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.  ... 
arXiv:0912.5279v1 fatcat:45de4mptpbdeveyf4x67ssuzr4

Biquadratic reciprocity and a Lucasian primality test

Pedro Berrizbeitia, T. G. Berry
2003 Mathematics of Computation  
Then, for a suitable seed s 0 , 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  ... 
doi:10.1090/s0025-5718-03-01575-8 fatcat:i6xesklpvfbx7mwsjcs7zxcl5e

Chebyshev polynomials and higher order Lucas Lehmer algorithm [article]

Kok Seng Chua
2021 arXiv   pre-print
We extend 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.  ... 
arXiv:2010.02677v2 fatcat:j3cqze2r6nb2znu7ldkjoru5ra

An elliptic curve test for Mersenne primes

Benedict H. Gross
2005 Journal of Number Theory  
Let 3 be a 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.  ... 
doi:10.1016/j.jnt.2003.11.011 fatcat:tdqa2so5yjfibkppmlpaifgxhq
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