# Gaussian Mersenne

Recall that the Mersenne primes are the primes of the form 2* ^{n}*-1.
There are no primes of the form

*b*

^{n}-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 *b ^{n}*-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

*b*-1 is -1, so there are no primes here! Finally if

^{n}*b*-1 = ±

*i*, then we get the conjugate pairs of numbers (1 ±

*i*)

*-1 with norms*

^{n}

_{}

and these can be prime!

It is easy to show that a Gaussian integer *a*+*b*i
is a Gaussian prime if and only if its norm

N(a+bi) =a^{2}+b^{2}

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)- 1 is Gaussian Mersenne prime if and only if^{n}nis 2, ornis odd and the norm is a rational prime.

These norms have been repeatedly studied as part
of the effort to factor 2^{n}± 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
2^{n}+1 and finally found the factorization of 2^{58}+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

-i((1+i)^{k}-1)

(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)

- A007671 Sloan's sequence A007671
- A007670 Sloan's sequence A007670
- Gaussian Mersenne's on the list of Largest Known Primes

**References:**

- BLS75
J. Brillhart,D. H. LehmerandJ. L. Selfridge, "New primality criteria and factorizations of 2^{m}± 1,"Math. Comp.,29(1975) 620--647.MR 52:5546[Thearticle for the classical (n^{2}-1) primality tests. Table errata in [Brillhart1982]]- BLSTW88
J. Brillhart,D. H. Lehmer,J. L. Selfridge,B. TuckermanandS. S. Wagstaff, Jr.,Factorizations of, Amer. Math. Soc., Providence RI, 1988. pp. xcvi+236, ISBN 0-8218-5078-4.b^{n}± 1,b=2,3,5,6,7,10,12 up to high powersMR 90d:11009(Annotation available)- CW25
A. J. C. CunninghamandH. J. Woodall,Factorizations of, Hodgson, 1925. London,y^{n}1, y =2,3,5,6,7,10,11,12 up to high powers (n)- HS76
M. HausmannandH. Shapiro, "Perfect ideals over the gaussian integers,"Comm. Pure Appl. Math.,29:3 (1976) 323--341.MR 54:12704- KR98a
R. KumanduriandC. 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- Spira61
R. Spira, "The complex sum of divisors,"Amer. Math. Monthly,68(1961) 120--124.MR 26:6101