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

%I A341789 #12 Feb 20 2021 07:56:41
%S A341789 4,9,17,19,23,25,29,37,47,49,59,67,71,73,83,89,103,107,121,127,131,
%T A341789 149,151,157,163,167,169,173,181,193,199,211,223,227,241,257,263,269,
%U A341789 277,283,293,307,317,349,359,389,397,419,421,431,439,449,457,461
%N A341789 Norms of prime elements in Z[(1+sqrt(-67))/2], the ring of integers of Q(sqrt(-67)).
%C A341789 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.
%C A341789 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.
%C A341789 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.
%H A341789 Jianing Song, <a href="/A341789/b341789.txt">Table of n, a(n) for n = 1..10000</a>
%e A341789 norm((1 + sqrt(-67))/2) = norm((1 - sqrt(-67))/2) = 17;
%e A341789 norm((3 + sqrt(-67))/2) = norm((3 - sqrt(-67))/2) = 19;
%e A341789 norm((5 + sqrt(-67))/2) = norm((5 - sqrt(-67))/2) = 23;
%e A341789 norm((7 + sqrt(-67))/2) = norm((7 - sqrt(-67))/2) = 29;
%e A341789 ...
%e A341789 norm((31 + sqrt(-67))/2) = norm((31 - sqrt(-67))/2) = 257.
%o A341789 (PARI) isA341783(n) = my(disc=-67); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)
%Y A341789 Cf. A011596, A106933, A191041, A191077.
%Y A341789 The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A318982.
%Y A341789 The total number of elements with norm n is given by A318984.
%Y A341789 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.
%K A341789 nonn,easy
%O A341789 1,1
%A A341789 _Jianing Song_, Feb 19 2021