A232931 The least positive integer k such that Kronecker(D/k) = -1 where D runs through all positive fundamental discriminants (A003658).

2, 3, 5, 2, 3, 2, 7, 5, 2, 5, 2, 7, 3, 3, 2, 3, 5, 13, 2, 3, 2, 5, 7, 2, 2, 5, 3, 3, 2, 5, 2, 3, 11, 2, 3, 11, 7, 7, 2, 7, 3, 3, 2, 7, 2, 3, 11, 2, 3, 2, 5, 5, 2, 5, 2, 11, 3, 3, 5, 2, 7, 11, 2, 3, 2, 5, 7, 2, 2, 5, 3, 3, 2, 7, 3, 11, 2, 3, 7, 7, 5, 2, 5, 2, 13, 3, 3, 2, 2, 3, 2, 3, 2, 5, 5, 11, 2, 7, 5, 3, 3, 5, 2, 3, 13, 5, 2, 3, 2, 17, 2, 2, 7, 3, 3, 2, 13, 2, 5, 2, 3, 5, 7, 5, 2, 5, 2, 11, 3, 2, 5, 2, 3, 7, 2, 3, 2, 17, 5, 7, 2, 7, 2, 5, 3, 3, 7, 2, 3, 7, 5, 2, 3

Offset: 2

Author

Steven Finch, Dec 02 2013

Keywords

• nonn

Comments

From Jianing Song, Jan 30 2019: (Start)

a(n) is necessarily prime. Otherwise, if a(n) is not prime, we have (D/p) = 0 or 1 for all prime divisors p of a(n), so (D/a(n)) must be 0 or 1 too, a contradiction.

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

Examples

			A003658(3) = 8, (8/3) = -1 and (8/2) = 0, so a(3) = 3.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 2..10000
• S. R. Finch, Average least nonresidues, December 4, 2013. [Cached copy, with permission of the author]
• Andrew Granville, R. A. Mollin and H. C. Williams, An upper bound on the least inert prime in a real quadratic field, Canad. J. Math. 52:2 (2000), pp. 369-380.
• P. Pollack, The average least quadratic nonresidue modulo m and other variations on a theme of Erdős, J. Number Theory 132 (2012) 1185-1202.
• Enrique Treviño, The least inert prime in a real quadratic field, Mathematics of Computation 81:279 (2012), pp. 1777-1797. See also his PANTS 2010 talk.

Crossrefs

Cf. A003658, A092419, A241482.

Programs

Mathematica

nMax = 200; A003658 = Select[Range[4nMax], NumberFieldDiscriminant[Sqrt[#]] == #&]; f[d_] := For[k = 1, True, k++, If[FreeQ[{0, 1}, KroneckerSymbol[d, k]], Return[k]]]; a[n_] := f[A003658[[n]]]; Table[a[n], {n, 2, nMax}] (* Jean-François Alcover, Nov 05 2016 *)

PARI

lp(D)=forprime(p=2,,if(kronecker(D,p)<0,return(p)))
for(n=5,1e3,if(isfundamental(n),print1(lp(n)", "))) \\ Charles R Greathouse IV, Apr 23 2014

Formula

With D = A003658(n): Mollin conjectured, and Granville, Mollin, & Williams proved, that for n > 1128, a(n) <= D^0.5 / 2. Treviño proves that for n > 484, a(n) <= D^0.45. Asymptotically the best known upper bound for the exponent is less than 0.16 when D is prime and 1/4 + epsilon (for any epsilon > 0) for general D. - Charles R Greathouse IV, Apr 23 2014 (corrected by Enrique Treviño, Mar 18 2022)

a(n) = A092419(A003658(n) - floor(sqrt(A003658(n)))), n >= 2. - Jianing Song, Jan 30 2019

Extensions

Name simplified by Jianing Song, Jan 30 2019