OFFSET
1,1
COMMENTS
2*sqrt(d)/Pi is the so-called "Minkowski's bound" for imaginary quadratic field. For other discriminants -d, there exists a prime p < 2*sqrt(d)/Pi such that Kronecker(-d,p) = 1.
Let K = Q(sqrt(-d)) be an imaginary quadratic field. The ideal class group of O_K (the ring of integers over K) is generated by ideal classes that contain I, where each I divides p*O_K for some p < 2*sqrt(d)/Pi. Note that if Kronecker(-d,p) = -1 (i.e., p is inert in K), then p*O_K is a prime ideal; if p | -d (i.e., p ramifies in K), then p*O_K = I^2, so the order of the ideal class that contains I is <= 2 in the ideal class group. So the ideal class group of Q(sqrt(-d)) necessarily has exponent <= 2 (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.). So this is a subsequence of A003644.
But there are other d such that the ideal class group of O_K has exponent 2. In fact, the exponent is <= 2 if and only if: for all primes p < 2*sqrt(d)/Pi, either (a) Kronecker(-d,p) = 0 or -1, or (b) Kronecker(-d,p) = 1, and 4*p^2 - d is a square. Here 2*sqrt(d)/Pi can be replaced by sqrt(d); conjecturally, if 2*sqrt(d)/Pi is replaced by sqrt(d/4), we get exactly the sequence A003644.
LINKS
Wikipedia, Minkowski's bound
EXAMPLE
For d = 708, the primes below 2*sqrt(708)/Pi ~ 16.94 are 2, 3, 5, 7, 11 and 13. We have 2, 3 | -708, Kronecker(-708,5) = Kronecker(-708,7) = Kronecker(-708,11) = Kronecker(-708,13) = -1, so 708 is a term.
PROG
(PARI) isA330221(d) = (d>0) && isfundamental(-d) && !sum(p=2, 2*sqrt(d)/Pi, isprime(p)&&kronecker(-d, p)==1)
CROSSREFS
KEYWORD
nonn,fini,more
AUTHOR
Jianing Song, Dec 06 2019
STATUS
approved