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%I #24 Apr 03 2023 10:36:12
%S 5,13,41,113,2113,525313,536903681,140737471578113,
%T 9444732965601851473921,604462909806215075725313,
%U 10384593717069655112945804582584321,2854495385411919762116496381035264358442074113
%N Gaussian-Mersenne primes: primes of the form ((1+i)^p - 1)((1-i)^p - 1).
%C See A057429 for the values of p.
%C Primes of the form q = 2^p +- 2^((p+1)/2) + 1. Note that q == 1 (mod p). - _Thomas Ordowski_, Apr 18 2019
%D John Brillhart et al., Factorizations of b^n +/- 1, b=2,3,5,6,7,10,12 up to high powers, Amer. Math. Soc., Providence RI, 1988, pp. xcvi+236.
%D R. K. Guy, Unsolved Problems in Number Theory, New York: Springer-Verlag, 1994, pp. 33-36.
%D Miriam Hausmann and Harold N. Shapiro, Perfect Ideals over the Gaussian Integers, Comm. Pure Appl. Math. 29 (1976), pp. 323-341.
%H Arkadiusz Wesolowski, <a href="/A182300/b182300.txt">Table of n, a(n) for n = 1..25</a>
%H Bogley, William A.; Williams, <a href="https://doi.org/10.1007/s00209-016-1664-3">Gerald Efficient finite groups arising in the study of relative asphericity</a>. Math. Z. 284, No. 1-2, 507-535 (2016).
%H Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/GaussianMersenne.html">Gaussian Mersenne</a>
%H C. K. Caldwell, "Top Twenty" page, <a href="https://t5k.org/top20/page.php?id=41">Gaussian Mersenne norm</a>
%H Ellen Gethner, Stan Wagon, and Brian Wick, <a href="http://mathdl.maa.org/images/upload_library/22/Chauvenet/Gethner.pdf">A Stroll Through the Gaussian Primes</a>, Amer. Math. Monthly 105 (1998), pp. 327-337.
%H W. L. McDaniel, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa25/aa2523.pdf">Perfect Gaussian integers</a>, Acta Arithmetica 25 (1974), pp. 137-144.
%H <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>
%t lst = {}; Do[a = (1 + I)^n - 1; b = a*Conjugate[a]; If[PrimeQ[b], AppendTo[lst, b]], {n, 151}]; lst
%t gmp[n_]:=Module[{x=(1+I)^n-1},x*Conjugate[x]]; Select[Table[gmp[n],{n,200}],PrimeQ] (* _Harvey P. Dale_, Apr 27 2016 *)
%Y Cf. A088962, A057429.
%K nice,nonn
%O 1,1
%A _Arkadiusz Wesolowski_, Apr 23 2012
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