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A341786 Norms of prime ideals in Z[(1+sqrt(-15))/2], the ring of integers of Q(sqrt(-15)).

%I A341786 #10 Feb 20 2021 11:21:31
%S A341786 2,3,5,17,19,23,31,47,49,53,61,79,83,107,109,113,121,137,139,151,167,
%T A341786 169,173,181,197,199,211,227,229,233,241,257,263,271,293,317,331,347,
%U A341786 349,353,379,383,409,421,439,443,467,499,503,541,557,563,571,587
%N A341786 Norms of prime ideals in Z[(1+sqrt(-15))/2], the ring of integers of Q(sqrt(-15)).
%C A341786 The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.
%C A341786 Note that Z[(1+sqrt(-15))/2] has class number 2.
%C A341786 Consists of the primes congruent to 1, 2, 3, 4, 5, 8 modulo 15 and the squares of primes congruent to 7, 11, 13, 14 modulo 15.
%C A341786 For primes p == 1, 4 (mod 15), there are two distinct ideals with norm p in Z[(1+sqrt(-15))/2], namely (x + y*(1+sqrt(-15))/2) and (x + y*(1-sqrt(-15))/2), where (x,y) is a solution to x^2 + x*y + 4*y^2 = p; for p == 2, 8 (mod 15), there are also two distinct ideals with norm p, namely (p, x + y*(1+sqrt(-15))/2) and (p, x + y*(1-sqrt(-15))/2), where (x,y) is a solution to x^2 + x*y + 4*y^2 = p^2 with y != 0; (3, sqrt(-15)) and (5, sqrt(-15)) are respectively the unique ideal with norm 3 and 5; for p == 7, 11, 13, 14 (mod 15), (p) is the only ideal with norm p^2.
%H A341786 Jianing Song, <a href="/A341786/b341786.txt">Table of n, a(n) for n = 1..10000</a>
%e A341786 Let |I| be the norm of an ideal I, then:
%e A341786 |(2, (1+sqrt(-15))/2)| = |(2, (1-sqrt(-15))/2)| = 2;
%e A341786 |(3, sqrt(-15))| = 3;
%e A341786 |(5, sqrt(-15))| = 5;
%e A341786 |(17, 7+4*sqrt(-15))| = |(17, 7-4*sqrt(-15))| = 17;
%e A341786 |(2 + sqrt(-15))| = |(2 - sqrt(-15))| = 19;
%e A341786 |(23, 17+4*sqrt(-15))| = |(23, 17-4*sqrt(-15))| = 23;
%e A341786 |(4 + sqrt(-15))| = |(4 - sqrt(-15))| = 31.
%o A341786 (PARI) isA341786(n) = my(disc=-15); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)
%Y A341786 Cf. A316569, A033212, A106859, A191018, A191062.
%Y A341786 The number of distinct ideals with norm n is given by A035175.
%Y A341786 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), this sequence (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 A341786 nonn,easy
%O A341786 1,1
%A A341786 _Jianing Song_, Feb 19 2021