Mersenne like Gaussian Primes

[Dean Hickerson]

Ed Pegg said:

> I think I discovered an analog to Gaussian-Mersenne primes
>
> http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?An
> um=057429
>
> 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,
> 85237,106693,160423,203789,364289
>
> These are primes of the form (1+i)^n - 1 .  (1+i)^4 - 1, -5,  is also prime.

Actually, it's not "primes of the form (1+i)^n - 1"; it's the values of n
for which (1+i)^n - 1 is prime.  And it depends on what you mean by prime:
If you're talking about Gaussian primes, then (1+i)^4 - 1 = -5 is not a
prime; it factors as (1+2i)(-1+2i).

It's well known that a Gaussian integer a+bi is a Gaussian prime if and
only if either its norm (which is the number times its conjugate,
(a+bi)(a-bi) = a^2+b^2) is an ordinary prime or a+bi is one of p, -p, ip,
or -ip where p is an ordinary prime that's congruent to 3 (mod 4).  For
numbers of the form (1+i)^n - 1, we can rule out the second case:
(1+i)^n-1 is pure imaginary iff n=1; it's real iff n is divisible by 4, in
which case it's divisible by 5.  So (1+i)^n - 1 is a Gaussian prime iff
(1+i)^n - 1 times its conjugate is an ordinary prime.  (The %N line of
A057429 uses this as the definition, thus avoiding the use of Gaussian
primes.  But it incorrectly says "primes of the form".)

> I took a look at primes of the form (1+i)^n + I, and thought it interesting
> enough to list.
>
> 2,3,4,6,7,8,11,14,16,19,38,47,62,79,151,163,167,214,239,254,
> 283,367,379,1214,1367,2558,4406

Again, these are the values of n, not the primes of the given form.  This
is now A027206, but its name line incorrectly refers to (1+i)^n + 1, not
(1+i)^n + i.  And the name line 'simplifies' this by saying that the
number times its conjugate is an ordinary prime.  In this case, that
simplification is not valid.  For example, if n=2, then (1+i)^2 + i = 3i
has norm 9, which is not prime, so 2 is not in the sequence with the
'simplified' definition.  In fact, all of the elements that are == 2 (mod 4)
are ruled out.

Incidentally, the corresponding question for the numbers (1+i)^n + 1 isn't
very interesting; they're only Gaussian primes for n=1, 2, 3, and 4.

I've sent corrected versions of these two sequences to Neil.

Dean Hickerson
dean at math.ucdavis.edu