A306537 The least prime q such that Kronecker(D/q) = 1 where D runs through all positive fundamental discriminants (A003658).

2, 11, 7, 11, 3, 2, 5, 5, 3, 5, 2, 3, 3, 2, 5, 7, 5, 2, 7, 3, 2, 5, 2, 3, 13, 3, 3, 2, 7, 7, 2, 5, 5, 2, 3, 2, 7, 3, 2, 3, 3, 2, 13, 5, 2, 5, 11, 5, 3, 2, 7, 11, 3, 13, 2, 3, 3, 2, 11, 2, 7, 2, 5, 3, 2, 11, 2, 3, 5, 3, 3, 2, 5, 13, 2, 13, 2, 3, 2, 5, 2, 3, 5, 2

Offset: 1

Author

Jianing Song, Feb 22 2019

Keywords

• nonn

Comments

a(n) is the least prime that decomposes in the real quadratic field with discriminant D, D = A003658(n).

For most n, a(n) is relatively small. There are only 459 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).

Also a(n) is the smallest prime p such that the real quadratic field with discriminant D = A003658(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021

Examples

			Let K = Q[sqrt(635)] with D = 2540 = A003658(774), we have: 2 and 5 divides 2540, (2540/3) = (2540/7) = ... = (2540/37) = -1 and (2540/41) = +1, so 2 and 5 ramify in K, 3, 7, ..., 37 remain inert in K and 41 decomposes in K, so a(774) = 41.
		

Crossrefs

Cf. A003658.

Similar sequences: A232931, A232932 (the least prime that remains inert); this sequence, A306538 (the least prime that decomposes); A306541, A306542 (the least prime that decomposes or ramifies).

Programs

PARI

b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))