A341788 Norms of prime elements in Z[(1+sqrt(-43))/2], the ring of integers of Q(sqrt(-43)).

4, 9, 11, 13, 17, 23, 25, 31, 41, 43, 47, 49, 53, 59, 67, 79, 83, 97, 101, 103, 107, 109, 127, 139, 167, 173, 181, 193, 197, 229, 239, 251, 269, 271, 281, 283, 293, 307, 311, 317, 337, 353, 359, 361, 367, 379, 397, 401, 431, 439, 443, 461, 479, 487, 509

Offset

1,1

Comments

Also norms of prime ideals in Z[(1+sqrt(-43))/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,43) >= 0 and the squares of primes such that (p,43) = -1, where (p,43) is the Legendre symbol.

For primes p such that (p,43) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-43))/2], namely (x + y*(1+sqrt(-43))/2) and (x + y*(1-sqrt(-43))/2), where (x,y) is a solution to x^2 + x*y + 11*y^2 = p; for p = 43, (sqrt(-43)) is the unique ideal with norm p; for primes p with (p,43) = -1, (p) is the only ideal with norm p^2.

Links

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

Example

norm((1 + sqrt(-43))/2) = norm((1 - sqrt(-43))/2) = 11;
norm((3 + sqrt(-43))/2) = norm((3 - sqrt(-43))/2) = 13;
norm((5 + sqrt(-43))/2) = norm((5 - sqrt(-43))/2) = 17;
norm((7 + sqrt(-43))/2) = norm((7 - sqrt(-43))/2) = 23;
...
norm((19 + sqrt(-43))/2) = norm((19 - sqrt(-43))/2) = 101.

Prog

(PARI) isA341788(n) = my(disc=-43); (isprime(n) && kronecker(disc, n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc, n)==-1)

Crossrefs

Cf. A011591, A106891, A191031, A184902.

The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035147.

The total number of elements with norm n is given by A138811.

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*), this sequence (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: A243651 A292673 A035233 * A010396 A010431 A010438

Adjacent sequences: A341785 A341786 A341787 * A341789 A341790 A341791

Keyword

nonn,easy

Author

Jianing Song, Feb 19 2021

Status

approved