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A341783 Absolute values of norms of prime elements in Z[(1+sqrt(5))/2], the ring of integers of Q(sqrt(5)). 9
4, 5, 9, 11, 19, 29, 31, 41, 49, 59, 61, 71, 79, 89, 101, 109, 131, 139, 149, 151, 169, 179, 181, 191, 199, 211, 229, 239, 241, 251, 269, 271, 281, 289, 311, 331, 349, 359, 379, 389, 401, 409, 419, 421, 431, 439, 449, 461, 479, 491, 499, 509, 521, 529 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Also norms of prime ideals in Z[(1+sqrt(5))/2], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.

Consists of the primes congruent to 0, 1, 4 modulo 5 and the squares of primes congruent to 2, 3 modulo 5.

For primes p == 1, 4 (mod 5), there are two distinct ideals with norm p in Z[(1+sqrt(5))/2], namely (x + y*(1+sqrt(5))/2) and (x + y*(1-sqrt(5))/2), where (x,y) is a solution to x^2 + x*y - y^2 = p; for p = 5, (sqrt(5)) is the unique ideal with norm p; for p == 2, 3 (mod 5), (p) is the only ideal with norm p^2.

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

EXAMPLE

norm((7 + sqrt(5))/2) = norm((7 - sqrt(5))/2) = 11;

norm((9 + sqrt(5))/2) = norm((9 - sqrt(5))/2) = 19;

norm((11 + sqrt(5))/2) = norm((11 - sqrt(5))/2) = 29;

norm(6 + sqrt(5)) = norm(6 - sqrt(5)) = 31.

PROG

(PARI) isA341783(n) = my(disc=5); (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1)

CROSSREFS

Cf. A080891, A038872, A045468, A003631.

The number of nonassociative elements with absolute value of norm n (also the number of distinct ideals with norm n) is given by A035187.

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), this sequence (D=5), A055664 (D=-3), A055025 (D=-4), A090348 (D=-7), A341784 (D=-8), A341785 (D=-11), A341786 (D=-15*), A341787 (D=-19), A091727 (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: A155149 A024821 A059610 * A319606 A230239 A194154

Adjacent sequences:  A341780 A341781 A341782 * A341784 A341785 A341786

KEYWORD

nonn,easy

AUTHOR

Jianing Song, Feb 19 2021

STATUS

approved

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Last modified September 23 11:45 EDT 2021. Contains 347612 sequences. (Running on oeis4.)