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A341784 Norms of prime elements in Z[sqrt(-2)], the ring of integers of Q(sqrt(-2)).

%I A341784 #13 Feb 20 2021 07:56:08
%S A341784 2,3,11,17,19,25,41,43,49,59,67,73,83,89,97,107,113,131,137,139,163,
%T A341784 169,179,193,211,227,233,241,251,257,281,283,307,313,331,337,347,353,
%U A341784 379,401,409,419,433,443,449,457,467,491,499,521,523,529,547,563
%N A341784 Norms of prime elements in Z[sqrt(-2)], the ring of integers of Q(sqrt(-2)).
%C A341784 Also norms of prime ideals in Z[sqrt(-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 A341784 Consists of the primes congruent to 1, 2, 3 modulo 8 and the squares of primes congruent to 5, 7 modulo 8.
%C A341784 For primes p == 1, 3 (mod 8), there are two distinct ideals with norm p in Z[sqrt(2)], namely (x + y*sqrt(-2)) and (x - y*sqrt(-2)), where (x,y) is a solution to x^2 + 2*y^2 = p; for p = 2, (sqrt(-2)) is the unique ideal with norm p; for p == 5, 7 (mod 8), (p) is the only ideal with norm p^2.
%H A341784 Jianing Song, <a href="/A341784/b341784.txt">Table of n, a(n) for n = 1..10000</a>
%e A341784 norm(1 + sqrt(-2)) = norm(1 + sqrt(-2)) = 3;
%e A341784 norm(3 + sqrt(-2)) = norm(3 + sqrt(-2)) = 11;
%e A341784 norm(3 + 2*sqrt(-2)) = norm(3 + 2*sqrt(-2)) = 17;
%e A341784 norm(1 + 3*sqrt(-2)) = norm(1 + 3*sqrt(-2)) = 19.
%o A341784 (PARI) isA341784(n) = my(disc=-8); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)
%Y A341784 Cf. A188510, A033203, A033200, A003628.
%Y A341784 The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A002325.
%Y A341784 The total number of elements with norm n is given by A033715.
%Y A341784 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), this sequence (D=-8), A341785 (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 A341784 nonn,easy
%O A341784 1,1
%A A341784 _Jianing Song_, Feb 19 2021