A118916 -id:A118916 - cp's OEIS frontend

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

N. J. A. Sloane, Jun 09 2000

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.

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

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

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

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 *)

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

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

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

A118917 Number of inequivalent primes in ring of integers Z[sqrt(2)] with absolute value of norm = n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0

Offset: 0

Franklin T. Adams-Watters, May 05 2006

Since there are infinitely many units in Z[sqrt(2)], the total number of primes with a given norm is infinite (when there are any).

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A055029, A118916.

a[n_] := Which[PrimeQ[n] && MatchQ[Mod[n, 8], 1|7], 2, p = Sqrt[n]; n == 2 || IntegerQ[p] && PrimeQ[p] && MatchQ[Mod[p, 8], 3|5], 1, True, 0]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 22 2016 *)

A118917(n) = { my(p); if(isprime(n)&&((1==(n%8))||(7==(n%8))), 2, if((2==n)||((issquare(n,&p)&&isprime(p))&&((3==(p%8))||(5==(p%8)))),1,0)); }; \\ Antti Karttunen, Aug 30 2017

a(n) = 2 if n is a prime = 1,7 (mod 8); a(n) = 1 if n is 2 or p^2 where p is a prime = 3,5 (mod 8); otherwise a(n) = 0.

cp's OEIS Frontend

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

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