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 A091727 Norms of prime ideals of Z[sqrt(-5)]. 13
 2, 3, 5, 7, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 103, 107, 109, 121, 127, 149, 163, 167, 169, 181, 223, 227, 229, 241, 263, 269, 281, 283, 289, 307, 347, 349, 361, 367, 383, 389, 401, 409, 421, 443, 449, 461, 463, 467, 487 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Consists of primes congruent to 1, 2, 3, 5, 7, 9 (mod 20) together with the squares of all other primes. From Jianing Song, Feb 20 2021: (Start) The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I. Note that Z[sqrt(-5)] has class number 2. 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. 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. For p == 11, 13, 17, 19 (mod 20), (p) is the only ideal with norm p^2. (End) REFERENCES David A. Cox, Primes of the form x^2+ny^2, Wiley, 1989. A. Frohlich and M. J. Taylor, Algebraic number theory, Cambridge university press, 1991. LINKS Jianing Song, Table of n, a(n) for n = 1..10000 EXAMPLE From Jianing Song, Feb 20 2021: (Start) Let |I| be the norm of an ideal I, then: |(2, 1+sqrt(-5))| = 2; |(3, 2+sqrt(-5))| = |(3, 2-sqrt(-5))| = 3; |(sqrt(-5))| = 5; |(7, 1+3*sqrt(-5))| = |(7, 1-3*sqrt(-5))| = 7; |(23, 22+3*sqrt(-5))| = |(23, 22-3*sqrt(-5))| = 23; |(3 + 2*sqrt(-5))| = |(3 - 2*sqrt(-5))| = 29; |(6 + sqrt(-5))| = |(6 - sqrt(-5))| = 41. (End) PROG (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 CROSSREFS Cf. A091728. Cf. A289741, A033205, A106865, A139513, A003626. The number of distinct ideals with norm n is given by A035170. 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. Sequence in context: A024770 A038603 A106116 * A240920 A030144 A343590 Adjacent sequences: A091724 A091725 A091726 * A091728 A091729 A091730 KEYWORD easy,nonn AUTHOR Paul Boddington, Feb 02 2004 EXTENSIONS Offset corrected by Jianing Song, Feb 20 2021 STATUS approved

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Last modified June 24 00:02 EDT 2024. Contains 373661 sequences. (Running on oeis4.)