A330221 Numbers d such that -d is a fundamental discriminant and all primes smaller than 2*sqrt(d)/Pi ramify or remain inert in the ring of integers of Q(sqrt(-d)).

3, 4, 7, 8, 11, 19, 20, 24, 40, 43, 51, 52, 67, 88, 115, 120, 123, 148, 163, 168, 228, 232, 235, 267, 280, 312, 372, 408, 427, 520, 708, 760, 840, 1320, 1848

Offset: 1

Author

Jianing Song, Dec 06 2019

Keywords

• nonn
• fini
• more

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.

Examples

			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.
		

Links

• Wikipedia, Minkowski's bound

Crossrefs

Cf. A003644.

Programs

PARI

isA330221(d) = (d>0) && isfundamental(-d) && !sum(p=2, 2*sqrt(d)/Pi, isprime(p)&&kronecker(-d,p)==1)