A182300 Gaussian-Mersenne primes: primes of the form ((1+i)^p - 1)((1-i)^p - 1).

5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, 9444732965601851473921, 604462909806215075725313, 10384593717069655112945804582584321, 2854495385411919762116496381035264358442074113

Offset: 1

Author

Arkadiusz Wesolowski, Apr 23 2012

Keywords

• nice
• nonn

Comments

See A057429 for the values of p.

Primes of the form q = 2^p +- 2^((p+1)/2) + 1. Note that q == 1 (mod p). - Thomas Ordowski, Apr 18 2019

References

• 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.
• R. K. Guy, Unsolved Problems in Number Theory, New York: Springer-Verlag, 1994, pp. 33-36.
• Miriam Hausmann and Harold N. Shapiro, Perfect Ideals over the Gaussian Integers, Comm. Pure Appl. Math. 29 (1976), pp. 323-341.

Links

• Arkadiusz Wesolowski, Table of n, a(n) for n = 1..25
• Bogley, William A.; Williams, Gerald Efficient finite groups arising in the study of relative asphericity. Math. Z. 284, No. 1-2, 507-535 (2016).
• Chris Caldwell, The Prime Glossary, Gaussian Mersenne
• C. K. Caldwell, "Top Twenty" page, Gaussian Mersenne norm
• Ellen Gethner, Stan Wagon, and Brian Wick, A Stroll Through the Gaussian Primes, Amer. Math. Monthly 105 (1998), pp. 327-337.
• W. L. McDaniel, Perfect Gaussian integers, Acta Arithmetica 25 (1974), pp. 137-144.
• Index entries for Gaussian integers and primes

Crossrefs

Cf. A088962, A057429.

Programs

Mathematica

lst = {}; Do[a = (1 + I)^n - 1; b = a*Conjugate[a]; If[PrimeQ[b], AppendTo[lst, b]], {n, 151}]; lst
gmp[n_]:=Module[{x=(1+I)^n-1},x*Conjugate[x]]; Select[Table[gmp[n],{n,200}],PrimeQ] (* Harvey P. Dale, Apr 27 2016 *)