A055673 Absolute values of norms of primes in ring of integers Z[sqrt(2)].

2, 7, 9, 17, 23, 25, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 121, 127, 137, 151, 167, 169, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 361, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593

Offset

1,1

Comments

The integers have the form z = a + b*sqrt(2), a and b rational integers. The norm of z is a^2 - 2*b^2, which may be negative.

References

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VII.

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Formula

Consists of 2; rational primes = +-1 (mod 8); and squares of rational primes = +-3 (mod 8).

Mathematica

maxNorm = 593; s1 = Select[Range[-1, maxNorm, 8], PrimeQ]; s2 = Select[Range[1, maxNorm, 8], PrimeQ]; s3 = Select[Range[-3, Sqrt[maxNorm], 8], PrimeQ]^2; s4 = Select[Range[3, Sqrt[maxNorm], 8], PrimeQ]^2; Union[{2}, s1, s2, s3, s4] (* Jean-François Alcover, Dec 07 2012, from formula *)

Prog

(PARI) is(n)=!!if(isprime(n), setsearch([1, 2, 7], n%8), issquare(n, &n) && isprime(n) && setsearch([3, 5], n%8)) \\ Charles R Greathouse IV, Sep 10 2016

Crossrefs

Cf. A055026-A055029, A055669-A055672, A118916, A118917.

Sequence in context: A005988 A199537 A079326 * A177737 A336770 A361706

Adjacent sequences: A055670 A055671 A055672 * A055674 A055675 A055676

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jun 09 2000

Extensions

I would also like to get the sequences (analogous to A055027 and A055029) giving the number of inequivalent primes mod units. Of course now there are infinitely many units.

More terms from Franklin T. Adams-Watters, May 05 2006

Status

approved