Gaussian Mersenne

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).

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

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.

In 1961, R. Spira defined the notion a sum of divisor function in the ring of Gaussian integers [Spira61]. He used this to define the Gaussian Mersenne primes as the primes of the form


(associates of the term we define to be a Gaussian Mersenne). W. L. McDaniel [McDaniel73] showed these numbers share many properties with the original Mersenne primes and perfect numbers. In 1976, Hausman and Shapiro started with a more natural definition of perfect numbers (ideals) [HS76].

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.

Related pages (outside of this work)


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]]
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., 1988.  Providence RI, pp. xcvi+236, ISBN 0-8218-5078-4. MR 90d:11009 (Annotation available)
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.
M. Hausmann and H. Shapiro, "Perfect ideals over the gaussian integers," Comm. Pure Appl. Math., 29:3 (1976) 323--341.  MR 54:12704
R. Kumanduri and C. Romero, Number theory with computer applications, Prentice Hall, 1998.  Upper Saddle River, New Jersey,
W. McDaniel, "Perfect Gaussian integers," Acta. Arith., 25 (1973/74) 137--144.  MR 48:11034
R. Spira, "The complex sum of divisors," Amer. Math. Monthly, 68 (1961) 120--124.  MR 26:6101
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