|
%I #12 Feb 20 2021 07:56:14
%S 3,4,5,11,23,31,37,47,49,53,59,67,71,89,97,103,113,137,157,163,169,
%T 179,181,191,199,223,229,251,257,269,289,311,313,317,331,353,361,367,
%U 379,383,389,397,401,419,421,433,443,449,463,467,487,499,509,521
%N Norms of prime elements in Z[(1+sqrt(-11))/2], the ring of integers of Q(sqrt(-11)).
%C Also norms of prime ideals in Z[(1+sqrt(-11))/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.
%C Consists of the primes congruent to 0, 1, 3, 4, 5, 9 modulo 11 and the squares of primes congruent to 2, 6, 7, 8, 10 modulo 5.
%C For primes p == 1, 3, 4, 5, 9 (mod 11), there are two distinct ideals with norm p in Z[(1+sqrt(-11))/2], namely (x + y*(1+sqrt(-11))/2) and (x + y*(1-sqrt(-11))/2), where (x,y) is a solution to x^2 + x*y + 3*y^2 = p; for p = 11, (sqrt(-11)) is the unique ideal with norm p; for p == 2, 6, 7, 8, 10 (mod 11), (p) is the only ideal with norm p^2.
%H Jianing Song, <a href="/A341785/b341785.txt">Table of n, a(n) for n = 1..10000</a>
%e norm((1 + sqrt(-11))/2) = norm((1 - sqrt(-11))/2) = 3;
%e norm((3 + sqrt(-11))/2) = norm((3 - sqrt(-11))/2) = 5;
%e norm((9 + sqrt(-11))/2) = norm((9 - sqrt(-11))/2) = 23;
%e norm((5 + 3*sqrt(-11))/2) = norm((5 - 3*sqrt(-11))/2) = 31.
%o (PARI) isA341785(n) = my(disc=-11); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)
%Y Cf. A011582, A056874, A296920, A191060.
%Y The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035179.
%Y The total number of elements with norm n is given by A028609.
%Y 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), this sequence (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.
%K nonn,easy
%O 1,1
%A _Jianing Song_, Feb 19 2021
|
