A055025 Norms of Gaussian primes.

2, 5, 9, 13, 17, 29, 37, 41, 49, 53, 61, 73, 89, 97, 101, 109, 113, 121, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 361, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 529, 541, 557, 569

Offset: 1

Author

N. J. A. Sloane, Jun 09 2000

Keywords

• nonn
• easy
• nice

Comments

This is the range of the norm a^2 + b^2 of Gaussian primes a + b i. A239621 lists for each norm value a(n) one of the Gaussian primes as a, b with a >= b >= 0. In A239397, any of these (a, b) is followed by (b, a), except for a = b = 1. - Wolfdieter Lang, Mar 24 2014, edited by M. F. Hasler, Mar 09 2018

From Jean-Christophe Hervé, May 01 2013: (Start)

The present sequence is related to the square lattice, and to its division in square sublattices. Let's say that an integer n divides a lattice if there exists a sublattice of index n. Example: 2, 4, 5 divide the square lattice. Then A001481 (norms of Gaussian integers) is the sequence of divisors of the square lattice. Say that n is a "prime divisor" if the index-n sublattice is not contained in any other sublattice except the original lattice itself. The present sequence gives the "prime divisors" of the square lattice.

Similarly, A055664 (Norms of Eisenstein-Jacobi primes) is the sequence of "prime divisors" of the hexagonal lattice. (End)

The sequence is formed of 2, the prime numbers of form 4k+1, and the square of other primes (of form 4k+3). These are the primitive elements of A001481. With 0 and 1, they are the numbers that are uniquely decomposable in the sum of two squares. - Jean-Christophe Hervé, Nov 17 2013

Examples

			There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i). In A239621 2+i is listed as 2, 1.
		

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Eric Weisstein's World of Mathematics, Gaussian prime
• Wikipedia, Table of Gaussian integer factorizations
• Index entries for Gaussian integers and primes

Crossrefs

Cf. A055026, A055027, A055028, A055029, A055664-A055666, A001481.

Cf. A239397, A239621 (Gaussian primes).

Programs

Mathematica

Union[(#*Conjugate[#] & )[ Select[Flatten[Table[a + b*I, {a, 0, 23}, {b, 0, 23}]], PrimeQ[#, GaussianIntegers -> True] & ]]][[1 ;; 55]] (* Jean-François Alcover, Apr 08 2011 *)
(* Or, from formula: *) maxNorm = 569; s1 = Select[Range[1, maxNorm, 4], PrimeQ]; s3 = Select[Range[3, Sqrt[maxNorm], 4], PrimeQ]^2; Union[{2}, s1, s3]  (* Jean-François Alcover, Dec 07 2012 *)

PARI

list(lim)=my(v=List()); if(lim>=2, listput(v,2)); forprime(p=3,sqrtint(lim\1), if(p%4==3, listput(v,p^2))); forprime(p=5,lim, if(p%4==1, listput(v,p))); Set(v) \\ Charles R Greathouse IV, Feb 06 2017

PARI

isA055025(n)=(isprime(n) && n%4<3) || (issquare(n, &n) && isprime(n) && n%4==3) \\ Jianing Song, Aug 15 2023, based on Charles R Greathouse IV's program for A055664

Formula

Consists of 2; rational primes = 1 (mod 4) [A002144]; and squares of rational primes = 3 (mod 4) [A002145^2].

a(n) ~ 2n log n. - Charles R Greathouse IV, Feb 06 2017

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000