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Gaussian Mersenne norm |
The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.
Definitions and NotesBut what if we switch to the Gaussian integers, are there any Gaussian Mersenne primes? That is, are there any Gaussian primes of the form bn-1? If so, then b-1 would have to be a unit. The Gaussian integers have four units: 1, -1, i, and ±i. If b-1 is 1, then we get the usual Mersenne primes. If b-1 = -1, then bn-1 is -1, so there are no primes here! Finally if b-1 = ±i, then we get the conjugate pairs of numbers (1 ±i) n-1 with norms
and these can be prime!
It is easy to show that a Gaussian integer a+bi is a Gaussian prime if and only if its norm
N(a + bi) = a2+b2
is prime or b=0 and a is a prime congruent to 3 (mod 4). For example, the prime factors of two are 1+i and 1-i, both of which have norm 2. So we have the following result:
Theorem. (1-i)n-1 is Gaussian Mersenne prime if and only if n is 2, or n is odd and the norm
is a rational prime.
These norms have been repeatedly studied as part of the effort to factor 2n-1 because they occur as factors in Aurifeuillian factorization
24m-2 + 1 = (22m-1 + 2m + 1) (22m-1 - 2m + 1).
So the first 23 examples of Gaussian Mersennes norms can be found in table 2LM of [BLSTW88], 21 of these were known by the early 1960's. These correspond to the Gaussian Mersenne primes (1 - i)n-1 for the following values of n:
2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997.
Much earlier, the mathematician Landry devoted a good part of his life to factoring 2n+1 and finally found the factorization of 258+1 in 1869 (so he was essentially the first to find the Gaussian Mersenne with n=29). Just ten years later, Aurifeuille found the above factorization, which would have made Landry's massive effort trivial [KR98, p. 37]! In all the Cunningham project's papers and books, beginning with [CW25], these Gaussian Mersenne norms have assumed a major role.
Mike Oakes, who apparantly originated the approach we used above in the early 1970's, has recently extended the list of known Gaussian Mersennes dramatically. We now know (1-i)n-1 is prime for the following values of n:
2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423 and 203789.Gaussian Mersennes share many properties with the regular Mersennes and Oakes suggests they occur with the same density.
Record Primes of this Type
rank prime digits who when comment 1 215317227 + 27658614 + 1 4610945 L5123 Jul 2020 Gaussian Mersenne norm 41?, generalized unique 2 24792057 - 22396029 + 1 1442553 L3839 Apr 2014 Gaussian Mersenne norm 40?, generalized unique 3 23704053 + 21852027 + 1 1115032 L3839 Sep 2014 Gaussian Mersenne norm 39?, generalized unique 4 21667321 - 2833661 + 1 501914 L137 Jan 2011 Gaussian Mersenne norm 38?, generalized unique 5 21203793 - 2601897 + 1 362378 L192 Sep 2006 Gaussian Mersenne norm 37, generalized unique 6 2991961 - 2495981 + 1 298611 x28 Nov 2005 Gaussian Mersenne norm 36, generalized unique 7 2364289 - 2182145 + 1 109662 p58 Jun 2001 Gaussian Mersenne norm 35, generalized unique 8 2203789 + 2101895 + 1 61347 O Sep 2000 Gaussian Mersenne norm 34, generalized unique 9 2160423 - 280212 + 1 48293 O Sep 2000 Gaussian Mersenne norm 33, generalized unique 10 2106693 + 253347 + 1 32118 O Sep 2000 Gaussian Mersenne norm 32, generalized unique 11 285237 + 242619 + 1 25659 x16 Aug 2000 Gaussian Mersenne norm 31, generalized unique 12 277291 + 238646 + 1 23267 O Sep 2000 Gaussian Mersenne norm 30, generalized unique 13 249207 - 224604 + 1 14813 x16 Jul 2000 Gaussian Mersenne norm 29, generalized unique 14 227529 - 213765 + 1 8288 O Sep 2000 Gaussian Mersenne norm 28, generalized unique 15 214699 + 27350 + 1 4425 O Sep 2000 Gaussian Mersenne norm 27, generalized unique 16 210141 + 25071 + 1 3053 O Sep 2000 Gaussian Mersenne norm 26, generalized unique
References
- BLS75
- J. Brillhart, D. H. Lehmer and J. L. Selfridge, "New primality criteria and factorizations of 2m ± 1," Math. Comp., 29 (1975) 620--647. MR 52:5546 [The article for the classical (n2 -1) primality tests. Table errata in [Brillhart1982]]
- BLSTW88
- J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman and S. S. Wagstaff, Jr., Factorizations of bn ± 1, b=2,3,5,6,7,10,12 up to high powers, Amer. Math. Soc., Providence RI, 1988. pp. xcvi+236, ISBN 0-8218-5078-4. MR 90d:11009 (Annotation available)
- Chamberland2003
- Chamberland, Marc, "Binary BBP-formulae for logarithms and generalized Gaussian-Mersenne primes," Journal of Integer Sequences, 6:Article 03.3.7 (2003) 1--10.
- CW25
- A. J. C. Cunningham and H. J. Woodall, Factorizations of yn
1, y = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers (n), Hodgson, 1925. London,
- HS76
- M. Hausmann and H. Shapiro, "Perfect ideals over the gaussian integers," Comm. Pure Appl. Math., 29:3 (1976) 323--341. MR 54:12704
- KR98a
- R. Kumanduri and C. Romero, Number theory with computer applications, Prentice Hall, Upper Saddle River, New Jersey, 1998.
- McDaniel73
- W. McDaniel, "Perfect Gaussian integers," Acta. Arith., 25 (1973/74) 137--144. MR 48:11034
- PH2002
- Perschell, Karaloine and Huff, Loran, "Mersenne primes in imaginary quadratic number fields," (2002) avaliable from http://www.utm.edu/staff/caldwell/preprints/kpp/Paper2.pdf. (Abstract available)
- Spira61
- R. Spira, "The complex sum of divisors," Amer. Math. Monthly, 68 (1961) 120--124. MR 26:6101
