## Gaussian Mersenne norm |

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, 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 *b ^{n}*-1? If
so, then

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:

is a rational prime.

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

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

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.

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

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 1 2^{15317227}+ 2^{7658614}+ 14610945 L5123 Jul 2020 Gaussian Mersenne norm 41?, generalized unique 2 2^{4792057}- 2^{2396029}+ 11442553 L3839 Apr 2014 Gaussian Mersenne norm 40?, generalized unique 3 2^{3704053}+ 2^{1852027}+ 11115032 L3839 Sep 2014 Gaussian Mersenne norm 39?, generalized unique 4 2^{1667321}- 2^{833661}+ 1501914 L137 Jan 2011 Gaussian Mersenne norm 38?, generalized unique 5 2^{1203793}- 2^{601897}+ 1362378 L192 Sep 2006 Gaussian Mersenne norm 37, generalized unique 6 2^{991961}- 2^{495981}+ 1298611 x28 Nov 2005 Gaussian Mersenne norm 36, generalized unique 7 2^{364289}- 2^{182145}+ 1109662 p58 Jun 2001 Gaussian Mersenne norm 35, generalized unique 8 2^{203789}+ 2^{101895}+ 161347 O Sep 2000 Gaussian Mersenne norm 34, generalized unique 9 2^{160423}- 2^{80212}+ 148293 O Sep 2000 Gaussian Mersenne norm 33, generalized unique 10 2^{106693}+ 2^{53347}+ 132118 O Sep 2000 Gaussian Mersenne norm 32, generalized unique 11 2^{85237}+ 2^{42619}+ 125659 x16 Aug 2000 Gaussian Mersenne norm 31, generalized unique 12 2^{77291}+ 2^{38646}+ 123267 O Sep 2000 Gaussian Mersenne norm 30, generalized unique 13 2^{49207}- 2^{24604}+ 114813 x16 Jul 2000 Gaussian Mersenne norm 29, generalized unique 14 2^{27529}- 2^{13765}+ 18288 O Sep 2000 Gaussian Mersenne norm 28, generalized unique 15 2^{14699}+ 2^{7350}+ 14425 O Sep 2000 Gaussian Mersenne norm 27, generalized unique 16 2^{10141}+ 2^{5071}+ 13053 O Sep 2000 Gaussian Mersenne norm 26, generalized unique

- 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)- Chamberland2003
Chamberland, Marc, "Binary BBP-formulae for logarithms and generalized Gaussian-Mersenne primes,"Journal of Integer Sequences,6:Article 03.3.7 (2003) 1--10. http://www.emis.ams.org/journals/JIS/VOL6/Chamberland/chamberland60.pdf.- 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- PH2002
Perschell, KaraloineandHuff, Loran, "Mersenne primes in imaginary quadratic number fields," (2002) avaliable from http://www.utm.edu/staff/caldwell/preprints/kpp/Paper2.pdf. (Abstract available)- Spira61
R. Spira, "The complex sum of divisors,"Amer. Math. Monthly,68(1961) 120--124.MR 26:6101

Chris K. Caldwell
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