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

Author

N. J. A. Sloane, Jun 09 2000

Keywords

• nonn
• easy
• nice

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

Crossrefs

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

Programs

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 *)

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

Formula

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

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