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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) 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 b1. This is a problem because b1 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 b1 would have to be a unit. The Gaussian integers have four units: 1, 1, i, and i. If b1 is 1, then we get the usual Mersenne primes. If b1 = 1, then b^{n}1 is 1, so there are no primes here! Finally if b1 = + 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
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 1i, both of which have norm 2. So we have the following result:
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^{4m2} + 1 = (2^{2m1} + 2^{m} + 1) (2^{2m1}  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:
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.
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Chris K. Caldwell © 19992020 (all rights reserved)
