A341784 - OEIS

%I #13 Feb 20 2021 07:56:08

%S 2,3,11,17,19,25,41,43,49,59,67,73,83,89,97,107,113,131,137,139,163,

%T 169,179,193,211,227,233,241,251,257,281,283,307,313,331,337,347,353,

%U 379,401,409,419,433,443,449,457,467,491,499,521,523,529,547,563

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

%C 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 Consists of the primes congruent to 1, 2, 3 modulo 8 and the squares of primes congruent to 5, 7 modulo 8.

%C 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 Jianing Song, <a href="/A341784/b341784.txt">Table of n, a(n) for n = 1..10000</a>

%e norm(1 + sqrt(-2)) = norm(1 + sqrt(-2)) = 3;

%e norm(3 + sqrt(-2)) = norm(3 + sqrt(-2)) = 11;

%e norm(3 + 2*sqrt(-2)) = norm(3 + 2*sqrt(-2)) = 17;

%e norm(1 + 3*sqrt(-2)) = norm(1 + 3*sqrt(-2)) = 19.

%o (PARI) isA341784(n) = my(disc=-8); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

%Y Cf. A188510, A033203, A033200, A003628.

%Y The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A002325.

%Y The total number of elements with norm n is given by A033715.

%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), 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 nonn,easy

%O 1,1

%A _Jianing Song_, Feb 19 2021