# Mersenne: A new series of Mersenne-like Gaussian primes

```In 1969 I investigated the series of complex (Gaussian) integers:
s[n] = (1+i)^n - 1.
At that time I could only find the first few primes. Even now, there seem to
be no
references in the math literature or on the internet to this series, which
has so
many beautiful aspects.
It is best thought of as the complex counterpart of the Mersenne series.
As with the Mersennes, n must be prime else s[n] is composite.
For n = 2, s[n] = (1+i)^2 - 1 = 2*i - 1, with modulus (2*i-1)*(-2*i-1) = 5,
which is prime.
If n is odd, let n = 2*m+1, m >= 0. Then the modulus of s[n] is:
A[n] = ((1+i)^n-1)*((1-i)^n-1) = (2^m)^2 + (2^m + (-1)^(m*(m+1)/2))^2.
It is easy to prove that any prime factor of A[n] is of the form 4*k*n+1.
For n = 3, i.e. m = 1, A[n] = 2^2 + (2^2+(-1)^1)^2 = 4 + 9 = 13,
which is prime.
For odd n >= 5, i.e. m >= 2, I have proved that a necessary and sufficient
condition for A[n] to be prime is:
R[m]^m = -1 mod A[n],
where R[m] =  2^m-(-1)^((m*(m+1)/2)^2.
This is similar to the Lucas-Lehmer test, and can be programmed in just
the same way. The time for "mod A[n]" modulus reduction is only marginally
longer than in the Mersenne case, and is anyway negligible compared with
the time for FFT squaring.
Over the last year I have devoted 1 or 2 Intel PCs to the search for
primes of this form, and have to date found the first 33:-
Seq   n   A[n]                 Digits
1      2    5                            1
2      3    2^3+2^2+1               2
3      5    2^5+2^3+1               2
4      7    2^7-2^4+1               3
5     11    2^11+2^6+1              4
6     19    2^19+2^10+1             6
7     29    2^29+2^15+1             9
8     47    2^47-2^24+1            15
9     73    2^73-2^38+1            22
10     79   2^79-2^40+1            24
11    113   2^113-2^57+1           35
12    151   2^151-2^76+1           46
13    157   2^157+2^79+1           48
14    163   2^163+2^82+1           50
15    167   2^167-2^84+1           51
16    239   2^239-2^120+1          72
17    241   2^241-2^121+1          73
18    283   2^283+2^142+1          86
19    353   2^353-2^177+1         107
20    367   2^367-2^184+1         111
21    379   2^379+2^190+1         115
22    457   2^457-2^229+1         138
23    997   2^997+2^499+1         301
24   1367   2^1367-2^684+1        412
25   3041   2^3041-2^1521+1       916
26  10141   2^10141+2^5071+1    3,053
27  14699   2^14699+2^7350+1    4,425
28  27529   2^27529-2^13765+1   8,288
29  49207   2^49207-2^24604+1  14,813
30  77291   2^77291+2^38646+1  23,267
31  85237   2^85237+2^42619+1  25,659
32 106693   2^106693+2^53347+1 32,118
33 160423   2^160423-2^80212+1 48,293
The first 32 were found by my own implementation of the above algorithm; the
33rd
was shown as "PRP" on 22 August 2000 by a beta release of Chris Nash's superb
PrimeForm/GW, and then confirmed to be prime by my program.

The exponents increase in very much the same manner as the Mersenne ones.
The logic behind the conjecture by Wagstaff and others that the geometic mean
of
the ratio of successive terms should be approximated by 2^(e^(-gamma))
applies to
this series equally, and is about equally well exemplified, thereby enhancing
confidence in its validity. (A _proof_ of the conjecture, is, of course, far
out of reach.)

I hereby solicit the aid of those in GIMPS et al. to extend this series of
primes!

Regards

Mike Oakes
_________________________________________________________________________
Unsubscribe & list info -- http://www.scruz.net/~luke/signup.htm
Mersenne Prime FAQ      -- http://www.exu.ilstu.edu/mersenne/faq-mers.txt
```