cp's OEIS Frontend

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).

%I A330191 #17 Dec 07 2019 00:02:49
%S A330191 76,108,172,252,268,387,400,540,588,592,603,652,988,1068,1072,1332,
%T A330191 1467,2088,2608,2832,2907,3712,4075,5868
%N 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).
%C A330191 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.
%C A330191 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.
%C A330191 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.
%C A330191 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)).
%C A330191 We also have the following observations (not proved):
%C A330191 (a) if e = 2 (i.e., d is in A003171\A133675 = A133288), then b(-d) < d/4 unless d = 60;
%C A330191 (b) if e > 2, then b(-d) < sqrt(d/4) unless d is in this sequence.
%C A330191 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.
%C A330191 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.
%e A330191 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).
%e A330191 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).
%e A330191 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).
%o A330191 (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)
%Y A330191 Cf. A003171, A133675, A133288.
%K A330191 nonn,more
%O A330191 1,1
%A A330191 _Jianing Song_, Dec 04 2019