The Top Twenty: Gaussian Mersenne norm

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.

rank

prime

digits

who

when

comment

2^4792057 - 2^2396029 + 1

1442553

L3839

Apr 2014

Gaussian Mersenne norm 40?

2^3704053 + 2^1852027 + 1

1115032

L3839

Sep 2014

Gaussian Mersenne norm 39?

2^1667321 - 2⁸³³⁶⁶¹ + 1

501914

L137

Jan 2011

Gaussian Mersenne norm 38?

2^1203793 - 2⁶⁰¹⁸⁹⁷ + 1

362378

L192

Sep 2006

Gaussian Mersenne norm 37

2⁹⁹¹⁹⁶¹ - 2⁴⁹⁵⁹⁸¹ + 1

298611

x28

Nov 2005

Gaussian Mersenne norm 36

2³⁶⁴²⁸⁹ - 2¹⁸²¹⁴⁵ + 1

109662

p58

Jun 2001

Gaussian Mersenne norm 35

2²⁰³⁷⁸⁹ + 2¹⁰¹⁸⁹⁵ + 1

61347

Sep 2000

Gaussian Mersenne norm 34

2¹⁶⁰⁴²³ - 2⁸⁰²¹² + 1

48293

Sep 2000

Gaussian Mersenne norm 33

2¹⁰⁶⁶⁹³ + 2⁵³³⁴⁷ + 1

32118

Sep 2000

Gaussian Mersenne norm 32

2⁸⁵²³⁷ + 2⁴²⁶¹⁹ + 1

25659

x16

Aug 2000

Gaussian Mersenne norm 31

2⁷⁷²⁹¹ + 2³⁸⁶⁴⁶ + 1

23267

Sep 2000

Gaussian Mersenne norm 30

2⁴⁹²⁰⁷ - 2²⁴⁶⁰⁴ + 1

14813

x16

Jul 2000

Gaussian Mersenne norm 29

2²⁷⁵²⁹ - 2¹³⁷⁶⁵ + 1

8288

Sep 2000

Gaussian Mersenne norm 28

2¹⁴⁶⁹⁹ + 2⁷³⁵⁰ + 1

4425

Sep 2000

Gaussian Mersenne norm 27

2¹⁰¹⁴¹ + 2⁵⁰⁷¹ + 1

3053

Sep 2000

Gaussian Mersenne norm 26

Gaussian Mersenne norm

The Prime Pages

Definitions and Notes

Record Primes of this Type

References