cp's OEIS Frontend

A341788 Norms of prime elements in Z[(1+sqrt(-43))/2], the ring of integers of Q(sqrt(-43)).

%I A341788 #12 Feb 20 2021 07:56:35
%S A341788 4,9,11,13,17,23,25,31,41,43,47,49,53,59,67,79,83,97,101,103,107,109,
%T A341788 127,139,167,173,181,193,197,229,239,251,269,271,281,283,293,307,311,
%U A341788 317,337,353,359,361,367,379,397,401,431,439,443,461,479,487,509
%N A341788 Norms of prime elements in Z[(1+sqrt(-43))/2], the ring of integers of Q(sqrt(-43)).
%C A341788 Also norms of prime ideals in Z[(1+sqrt(-43))/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 A341788 Consists of the primes such that (p,43) >= 0 and the squares of primes such that (p,43) = -1, where (p,43) is the Legendre symbol.
%C A341788 For primes p such that (p,43) = 1, there are two distinct ideals with norm p in Z[(1+sqrt(-43))/2], namely (x + y*(1+sqrt(-43))/2) and (x + y*(1-sqrt(-43))/2), where (x,y) is a solution to x^2 + x*y + 11*y^2 = p; for p = 43, (sqrt(-43)) is the unique ideal with norm p; for primes p with (p,43) = -1, (p) is the only ideal with norm p^2.
%H A341788 Jianing Song, <a href="/A341788/b341788.txt">Table of n, a(n) for n = 1..10000</a>
%e A341788 norm((1 + sqrt(-43))/2) = norm((1 - sqrt(-43))/2) = 11;
%e A341788 norm((3 + sqrt(-43))/2) = norm((3 - sqrt(-43))/2) = 13;
%e A341788 norm((5 + sqrt(-43))/2) = norm((5 - sqrt(-43))/2) = 17;
%e A341788 norm((7 + sqrt(-43))/2) = norm((7 - sqrt(-43))/2) = 23;
%e A341788 ...
%e A341788 norm((19 + sqrt(-43))/2) = norm((19 - sqrt(-43))/2) = 101.
%o A341788 (PARI) isA341788(n) = my(disc=-43); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)
%Y A341788 Cf. A011591, A106891, A191031, A184902.
%Y A341788 The number of nonassociative elements with norm n (also the number of distinct ideals with norm n) is given by A035147.
%Y A341788 The total number of elements with norm n is given by A138811.
%Y A341788 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*), this sequence (D=-43), A341789 (D=-67), A341790 (D=-163). Here a "*" indicates the cases where O_K is not a unique factorization domain.
%K A341788 nonn,easy
%O A341788 1,1
%A A341788 _Jianing Song_, Feb 19 2021