A182300 - OEIS

A182300

Gaussian-Mersenne primes: primes of the form ((1+i)^p - 1)((1-i)^p - 1).

%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