A055027 Number of inequivalent Gaussian primes of successive norms (indexed by A055025).

1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2

Offset: 1

Author

N. J. A. Sloane, Jun 09 2000

Keywords

• nonn
• easy
• nice

Comments

These are the primes in the ring of integers a+bi, a and b rational integers, i = sqrt(-1).

Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-i).

Examples

			There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i).
		

References

• R. K. Guy, Unsolved Problems in Number Theory, A16.
• L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.

Links

• Index entries for Gaussian integers and primes

Crossrefs

Cf. A055025-A055029, A055664-...

Programs

Mathematica

norms = Union[ #*Conjugate[#]& [ Select[ Flatten[ Table[a + b*I, {a, 0, 31}, {b, 0, 31}]], PrimeQ[#, GaussianIntegers -> True] &]]]; f[norm_] := (Clear[a, b]; primes = {a + b*I} /. {ToRules[ Reduce[a^2 + b^2 == norm, {a, b}, Integers]]}; primes //. {p1___, p2_, p3___, p4_, p5___} /; MatchQ[p2, (-p4 | I*p4 | -I*p4)] :> {p1, p2, p3, p5} // Length); A055027 = f /@ norms (* Jean-François Alcover, Nov 30 2012 *)

Extensions

More terms from Reiner Martin, Jul 20 2001