A330191 For -d == 0, 1 (mod 4), let E(-d) to be the exponent of the class group of binary quadratic forms with discriminant -d, b(-d) to be the smallest prime p such that Kronecker(-d,p) = 1, then sequence gives d such that E(-d) > 2 and b(-d) > sqrt(d/4).

76, 108, 172, 252, 268, 387, 400, 540, 588, 592, 603, 652, 988, 1068, 1072, 1332, 1467, 2088, 2608, 2832, 2907, 3712, 4075, 5868

Offset: 1

Author

Jianing Song, Dec 04 2019

Keywords

• nonn
• more

Comments

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.

Examples

			76 is in this sequence because the class group of binary quadratic forms with discriminant -76 is isomorphic to C_3 (generated by 4x^2 - x*y + 5y^2), and the smallest prime p such that Kronecker(-76,p) = 1 is p = 5 > sqrt(76/4).
387 is in this sequence because the class group of binary quadratic forms with discriminant -387 is isomorphic to C_4 (generated by 9x^2 - 3x*y + 11y^2), and the smallest prime p such that Kronecker(-387,p) = 1 is p = 11 > sqrt(387/4).
5868 is in this sequence because the class group of binary quadratic forms with discriminant -5868 is isomorphic to C_12 (generated by 36x^2 - 6x*y + 41y^2), and the smallest prime p such that Kronecker(-5868,p) = 1 is p = 41 > sqrt(5868/4).
		

Crossrefs

Cf. A003171, A133675, A133288.

Programs

PARI

isok(d) = (d>0) && (-d)%4<=1 && (quadclassunit(-d)[2]!=[]&&quadclassunit(-d)[2][1]!=2) && !sum(p=1, sqrt(d/4), isprime(p)&&kronecker(-d,p)==1)