A341783 - OEIS

%I #13 Feb 20 2021 07:56:02

%S 4,5,9,11,19,29,31,41,49,59,61,71,79,89,101,109,131,139,149,151,169,

%T 179,181,191,199,211,229,239,241,251,269,271,281,289,311,331,349,359,

%U 379,389,401,409,419,421,431,439,449,461,479,491,499,509,521,529

%N Absolute values of norms of prime elements in Z[(1+sqrt(5))/2], the ring of integers of Q(sqrt(5)).

%C Also norms of prime ideals in Z[(1+sqrt(5))/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 0, 1, 4 modulo 5 and the squares of primes congruent to 2, 3 modulo 5.

%C For primes p == 1, 4 (mod 5), there are two distinct ideals with norm p in Z[(1+sqrt(5))/2], namely (x + y*(1+sqrt(5))/2) and (x + y*(1-sqrt(5))/2), where (x,y) is a solution to x^2 + x*y - y^2 = p; for p = 5, (sqrt(5)) is the unique ideal with norm p; for p == 2, 3 (mod 5), (p) is the only ideal with norm p^2.

%H Jianing Song, <a href="/A341783/b341783.txt">Table of n, a(n) for n = 1..10000</a>

%e norm((7 + sqrt(5))/2) = norm((7 - sqrt(5))/2) = 11;

%e norm((9 + sqrt(5))/2) = norm((9 - sqrt(5))/2) = 19;

%e norm((11 + sqrt(5))/2) = norm((11 - sqrt(5))/2) = 29;

%e norm(6 + sqrt(5)) = norm(6 - sqrt(5)) = 31.

%o (PARI) isA341783(n) = my(disc=5); (isprime(n) && kronecker(disc,n)>=0) || (issquare(n, &n) && isprime(n) && kronecker(disc,n)==-1)

%Y Cf. A080891, A038872, A045468, A003631.

%Y The number of nonassociative elements with absolute value of norm n (also the number of distinct ideals with norm n) is given by A035187.

%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), this sequence (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), 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