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A341790 Norms of prime elements in Z[(1+sqrt(-163))/2], the ring of integers of Q(sqrt(-163)).

%I A341790 #11 Feb 20 2021 07:56:47
%S A341790 4,9,25,41,43,47,49,53,61,71,83,97,113,121,131,151,163,167,169,173,
%T A341790 179,197,199,223,227,251,263,281,289,307,313,347,359,361,367,373,379,
%U A341790 383,397,409,419,421,439,457,461,487,499,503,523,529,547,563,577,593
%N A341790 Norms of prime elements in Z[(1+sqrt(-163))/2], the ring of integers of Q(sqrt(-163)).
%C A341790 Also norms of prime ideals in Z[(1+sqrt(-163))/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 A341790 Consists of the primes such that (p,163) >= 0 and the squares of primes such that (p,163) = -1, where (p,163) is the Legendre symbol.
%C A341790 For primes p such that (p,163) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-163))/2], namely (x + y*(1+sqrt(-163))/2) and (x + y*(1-sqrt(-163))/2), where (x,y) is a solution to x^2 + x*y + 41*y^2 = p; for p = 163, (sqrt(-163)) is the unique ideal with norm p; for primes p with (p,163) = -1, (p) is the only ideal with norm p^2.
%H A341790 Jianing Song, <a href="/A341790/b341790.txt">Table of n, a(n) for n = 1..10000</a>
%e A341790 norm((1 + sqrt(-163))/2) = norm((1 - sqrt(-163))/2) = 41;
%e A341790 norm((3 + sqrt(-163))/2) = norm((3 - sqrt(-163))/2) = 43;
%e A341790 norm((5 + sqrt(-163))/2) = norm((5 - sqrt(-163))/2) = 47;
%e A341790 norm((7 + sqrt(-163))/2) = norm((7 - sqrt(-163))/2) = 53;
%e A341790 ...
%e A341790 norm((79 + sqrt(-163))/2) = norm((79 - sqrt(-163))/2) = 1601.
%o A341790 (PARI) isA341783(n) = my(disc=-163); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)
%Y A341790 Cf. A011615, A257362, A296921, A296915.
%Y A341790 The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A318983.
%Y A341790 The total number of elements with norm n is given by A318985.
%Y A341790 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), A341789 (D=-67), this sequence (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.
%K A341790 nonn,easy
%O A341790 1,1
%A A341790 _Jianing Song_, Feb 19 2021