Gaussian Integers (A057429)

Franklin T. Adams-Watters franktaw at
Tue Feb 1 04:15:13 CET 2005

Hans Havermann <pxp at> wrote:

>> I found another interesting article about 'Mersenne Primes In Imaginary
>> Quadratic Number Fields' with Google. There are numbers (1+i)**n-1, 
>> which are
>> Gaussian primes. n has to be prime. The sequence in OEIS is A057429, 
>> found by
>> Mike Oakes and others.
>> Sequence:  
>> 2,3,5,7,11,19,29,47,73,79,113,151,157,163,167,239,241,283,...
>> Name:      Gaussian-Mersenne primes: (1+i)^n - 1 times its conjugate 
>> is prime.
>> Comments:  Equivalently, (1+i)^n - 1 is a Gaussian prime.
>Does the comment, I wonder, require a small caveat because (1+i)^4 - 1 
>is a Gaussian prime?
>PrimeQ[(1 + I)^4 - 1]

Ah, but (1+i)^4 - 1 = -5, which is an *integer* prime, but *not* a Gaussian prime.
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067

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