A341789 Norms of prime elements in Z[(1+sqrt(-67))/2], the ring of integers of Q(sqrt(-67)).

4, 9, 17, 19, 23, 25, 29, 37, 47, 49, 59, 67, 71, 73, 83, 89, 103, 107, 121, 127, 131, 149, 151, 157, 163, 167, 169, 173, 181, 193, 199, 211, 223, 227, 241, 257, 263, 269, 277, 283, 293, 307, 317, 349, 359, 389, 397, 419, 421, 431, 439, 449, 457, 461

Offset: 1

Author

Jianing Song, Feb 19 2021

Keywords

• nonn
• easy

Comments

Also norms of prime ideals in Z[(1+sqrt(-67))/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 such that (p,67) >= 0 and the squares of primes such that (p,67) = -1, where (p,67) is the Legendre symbol.

For primes p such that (p,67) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-67))/2], namely (x + y*(1+sqrt(-67))/2) and (x + y*(1-sqrt(-67))/2), where (x,y) is a solution to x^2 + x*y + 17*y^2 = p; for p = 67, (sqrt(-67)) is the unique ideal with norm p; for primes p with (p,67) = -1, (p) is the only ideal with norm p^2.

Examples

			norm((1 + sqrt(-67))/2) = norm((1 - sqrt(-67))/2) = 17;
norm((3 + sqrt(-67))/2) = norm((3 - sqrt(-67))/2) = 19;
norm((5 + sqrt(-67))/2) = norm((5 - sqrt(-67))/2) = 23;
norm((7 + sqrt(-67))/2) = norm((7 - sqrt(-67))/2) = 29;
...
norm((31 + sqrt(-67))/2) = norm((31 - sqrt(-67))/2) = 257.
		

Links

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

Crossrefs

Cf. A011596, A106933, A191041, A191077.

The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A318982.

The total number of elements with norm n is given by A318984.

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), A091727 (D=-20*), A341788 (D=-43), this sequence (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.

Programs

PARI

isA341783(n) = my(disc=-67); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)