A330162 For imaginary fundamental discriminants -d, define b(-d) to be the smallest prime p such that Kronecker(-d,p) = 1. Sequence gives d such that b(-d)^3 > d/4 > b(-d)^2.

23, 31, 56, 59, 68, 83, 104, 107, 136, 139, 184, 211, 219, 244, 259, 264, 276, 283, 291, 292, 307, 328, 331, 339, 355, 376, 379, 388, 411, 424, 436, 451, 456, 472, 499, 523, 547, 552, 568, 580, 628, 643, 667, 712, 723, 763, 772, 787, 808, 820, 835, 843, 852, 868, 883

Offset: 1

Author

Jianing Song, Dec 04 2019

Keywords

• nonn

Comments

It seems that this sequence contains 810 terms, the largest being 1154008. In general, it seems that for any t > 0, b(-d) = o(d^t) as -d -> -oo.

For fundamental discriminants -d, we want to determine the size of b(-d), i.e., the size of the smallest prime that decomposes in Q[sqrt(-d)].

Let K = Q[sqrt(-d)], O_K be the ring of integers over K, so O_K is a Dedekind domain. Let E(-d) be the exponent of the ideal class group of O_K (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).

If Kronecker(-d,p) = 1, it is well-known that p*O_K is the product of two distinct prime ideals of O_K, say, p*O_K = I*I'. By the property of the ideal class group of Q[sqrt(-d)], I^(k*e) must be principal, e = E(-d). Let t*O_K = I^(k*e), then t/p is not an algebraic integer, and the norm of t 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 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)).

If E(-d) = 3, then d is in this sequence.

We also have the following observations (not proved):

(a) if e = 2 (i.e., d is in A003644\A014602 = A316743), then b(-d) < d/4;

(b) if e > 2, then b(-d) < sqrt(d/4) (it can be proved by using deeper algebraic number theory that b(-d) < 2*sqrt(d)/Pi).

If these observations are true, this sequence is also the list of d such that b(-d) > (d/4)^(1/3) and d is not in A003644.

Note that 5460 is conjectured to be the largest term in A003644. Therefore, it seems that b(-d) < sqrt(d/4) for all d > 5460; it seems that b(-d) < (d/4)^(1/3) for all d > 1154008.

Among the known terms:

(1) the term d with the largest E(-d) is d = 998328 with E(-d) = 66.

(2) the term d with the largest b(-d) is d = 656755 with b(-d) = 79.

(3) the largest prime is d = 90787 with E(-d) = 23.

Examples

			The smallest prime p such that Kronecker(-499,p) = 1 is p = 5, and 5^3 > 499/4 > 5^2, so 499 is a term.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..810

Crossrefs

Cf. A014602, A003644, A316743, A306538.

Programs

PARI

b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
isA330162(d) = (d>0) && isfundamental(-d) && b(-d) > sqrtn(d/4,3) && b(-d) < sqrt(d/4)