Gaussian Mersenne norm

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Definitions and Notes

Recall that the Mersenne primes are the primes of the form 2ⁿ-1. There are no primes of the form bⁿ-1 for any other positive integer b because these numbers are all divisible by b-1. This is a problem because b-1 is not a unit (that is, it is not +1 or -1, the divisors of 1).

But 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

2^4m-2 + 1 = (2^2m-1 + 2^m + 1) (2^2m-1 - 2^m + 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 2^15317227 + 2^7658614 + 1 4610945 L5123 Jul 2020 Gaussian Mersenne norm 41?, generalized unique
2 2^4792057 - 2^2396029 + 1 1442553 L3839 Apr 2014 Gaussian Mersenne norm 40?, generalized unique
3 2^3704053 + 2^1852027 + 1 1115032 L3839 Sep 2014 Gaussian Mersenne norm 39?, generalized unique
4 2^1667321 - 2⁸³³⁶⁶¹ + 1 501914 L137 Jan 2011 Gaussian Mersenne norm 38?, generalized unique
5 2^1203793 - 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 2^m ± 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, London, 1925.
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, 1998. Upper Saddle River, New Jersey,
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