A091727 - OEIS

%I #14 Feb 20 2021 13:31:07

%S 2,3,5,7,23,29,41,43,47,61,67,83,89,101,103,107,109,121,127,149,163,

%T 167,169,181,223,227,229,241,263,269,281,283,289,307,347,349,361,367,

%U 383,389,401,409,421,443,449,461,463,467,487

%N Norms of prime ideals of Z[sqrt(-5)].

%C Consists of primes congruent to 1, 2, 3, 5, 7, 9 (mod 20) together with the squares of all other primes.

%C From _Jianing Song_, Feb 20 2021: (Start)

%C The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.

%C Note that Z[sqrt(-5)] has class number 2.

%C For primes p == 1, 9 (mod 20), there are two distinct ideals with norm p in Z[sqrt(-5)], namely (x + y*sqrt(-5)) and (x - y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p.

%C For p == 3, 7 (mod 20), there are also two distinct ideals with norm p, namely (p, x+y*sqrt(-5)) and (p, x-y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p^2 with y != 0; (2, 1+sqrt(-5)) and (sqrt(-5)) are respectively the unique ideal with norm 2 and 5.

%C For p == 11, 13, 17, 19 (mod 20), (p) is the only ideal with norm p^2. (End)

%D David A. Cox, Primes of the form x^2+ny^2, Wiley, 1989.

%D A. Frohlich and M. J. Taylor, Algebraic number theory, Cambridge university press, 1991.

%H Jianing Song, <a href="/A091727/b091727.txt">Table of n, a(n) for n = 1..10000</a>

%e From _Jianing Song_, Feb 20 2021: (Start)

%e Let |I| be the norm of an ideal I, then:

%e |(2, 1+sqrt(-5))| = 2;

%e |(3, 2+sqrt(-5))| = |(3, 2-sqrt(-5))| = 3;

%e |(sqrt(-5))| = 5;

%e |(7, 1+3*sqrt(-5))| = |(7, 1-3*sqrt(-5))| = 7;

%e |(23, 22+3*sqrt(-5))| = |(23, 22-3*sqrt(-5))| = 23;

%e |(3 + 2*sqrt(-5))| = |(3 - 2*sqrt(-5))| = 29;

%e |(6 + sqrt(-5))| = |(6 - sqrt(-5))| = 41. (End)

%o (PARI) isA091727(n) = { my(ms = [1, 2, 3, 5, 7, 9], p, e=isprimepower(n,&p)); if(!e || e>2, 0, bitxor(e-1,!!vecsearch(ms,p%20))); }; \\ _Antti Karttunen_, Feb 24 2020

%Y Cf. A091728.

%Y Cf. A289741, A033205, A106865, A139513, A003626.

%Y The number of distinct ideals with norm n is given by A035170.

%Y Norms of prime ideals in O_K, where K is the quadratic field with discriminant D and O_K be the ring of integers of K: A055673 (D=8), A341783 (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), this sequence (D=-20*), A341788 (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

%K easy,nonn

%O 1,1

%A _Paul Boddington_, Feb 02 2004

%E Offset corrected by _Jianing Song_, Feb 20 2021