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
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:
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
-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)