cp's OEIS Frontend

A003171 = 4*A000926 U {a(n)}.

Note that d is in A000926 (i.e., 4d is in A003171) if and only if: for all gcd(d,k) = 1, if k^2 < 3d, then d + k^2 is either a prime, or twice a prime, or the square of a prime, or 8 or 16. It seems that d is in this sequence if and only if: for all odd k, gcd(d,k) = 1, if k^2 < 3d, then (d + k^2)/4 is either a prime or the square of a prime.

It is conjectured that this is the full list. Otherwise, there could be at most one more term d such that -d is a fundamental discriminant.

k is a term iff the form class group of positive binary quadratic forms with discriminant -k is isomorphic to (C_2)^r X C_4.

This is a subsequence of A133676, so it's finite. It seems that this sequence has 324 terms, the largest being 87360.

The smallest number in A133676 but not here is 3600.

Discriminant = -760. See A139827 for more information.

10*x^2 + 19 produces 19 consecutive primes belonging to A028416 for x from 0 to 18. - Davide Rotondo, Jun 13 2022

Primes p such that Kronecker(2,p) <= 0, Kronecker(5,p) >= 0 and Kronecker(-19,p) <= 0. - Jianing Song, Jun 13 2022

The exponent of a group G is the smallest e > 0 such that x^e = I for all x in G, where I is the group identity.

d is in this sequence if and only -d == 0, 1 (mod 4), d is not in A003171 and b(-d) > sqrt(d/4). It seems that 5868 is the largest term. In general, it seems that for any t > 0, b(-d) = o(d^t) as -d -> -oo.

For -d == 0, 1 (mod 4), we want to determine the size of b(-d). Let R = Z[sqrt(-d)/2] if -d == 0 (mod 4), R = Z[(1+sqrt(-d))/2)] otherwise, note that R is not necessarily a Dedekind domain. It is conjectured that if Kronecker(-d,p) = 1, then p*R is the product of two distinct prime ideals of R (this is obviously true if -d is a fundamental discriminant). It also seems that if p*R = I*I', then I^(k*e) must be principal, e = E(-d) (again, this is true if -d is fundamental). If these statements are indeed true, let t*O_k = I^(k*e), then t/p is not in R, and the norm of t over R is p^e. Define f(x,y) = x^2 + (d/4)*y^2 if -d == 0 (mod 4), x^2 + x*y + ((d+1)/4)*y^2 otherwise, it is easy to see f(x,y) = p^(k*e) has integral solutions (x,y) such that gcd(x,y) = 1.

If f(x,y) = p^(k*e) < d, then |y| = 1, so it seems that 4*p^(k*e) - d must be a (positive) square. Setting k = 1 gives b(-d) > (d/4)^(1/e) (and furthermore we have: if Kronecker(-d,p) = 1 and p^(k*e) < d, then k = 1, or (p,k,e,d) = (2,2,1,7), (3,2,1,11)).

We also have the following observations (not proved):

(a) if e = 2 (i.e., d is in A003171\A133675 = A133288), then b(-d) < d/4 unless d = 60;

(b) if e > 2, then b(-d) < sqrt(d/4) unless d is in this sequence.

Note that 7392 is conjectured to be the largest term in A003171. Therefore, it seems that b(-d) < sqrt(d/4) for all d > 7392.

Although these terms satisfy b(-d) > sqrt(d/4), for each d it is fairly simple to find a prime p such that Kronecker(-d,p) = 1 and f(x,y) = p^2 has no coprime solution (x,y). In contrast, if (a) is true, for d in A003171 (i.e., d such that E(-d) <= 2) we have b(-d) > sqrt(d/4); if Kronecker(-d,p) = 1 then there always exists coprime (x,y) such that f(x,y) = p^2.