

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
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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.
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, xy*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



EXAMPLE

Let I be the norm of an ideal I, then:
(2, 1+sqrt(5)) = 2;
(3, 2+sqrt(5)) = (3, 2sqrt(5)) = 3;
(sqrt(5)) = 5;
(7, 1+3*sqrt(5)) = (7, 13*sqrt(5)) = 7;
(23, 22+3*sqrt(5)) = (23, 223*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(e1, !!vecsearch(ms, p%20))); }; \\ Antti Karttunen, Feb 24 2020


CROSSREFS

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.


KEYWORD

easy,nonn


AUTHOR



EXTENSIONS



STATUS

approved



