A394567 Product of the three cubic Gauss periods for primes q == 1 (mod 3).

1, -1, 7, 8, -11, -8, 9, -5, 27, -41, 79, 61, 4, -80, -103, 123, -64, 169, 67, -143, -59, 125, 256, 212, -125, -261, -236, -304, 216, -371, 49, -25, 517, -435, 221, -365, 544, 515, 343, 16, 504, 220, -343, 505, -536, 891, -521, 81, 719, -171, -512, 1169, -999, -321

Offset

1,3

Comments

a(n) is the product of the three cubic Gauss periods for the n-th prime q == 1 (mod 3).

References

B. C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jacobi Sums, Wiley-Interscience, 1998, Theorem 2.5.1.

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Springer GTM 84, 1990, Chapters 8-9.

Links

Dao Van Tam, Table of n, a(n) for n = 1..10000

Dao Van Tam, Lean 4 formal verification

Formula

a(n) = ((c + 3)*q - 1)/27 where q = A002476(n), x and y are integers with q = x^2 + x*y + 7*y^2 and gcd(x,y) = 1, and c = 2*x + y with c == 1 (mod 3).

a(n) = (L^3 - 3*q*L + 3*q - 1)/27 where 4*q = L^2 + 27*M^2, L == 1 (mod 3), with L = 2*x + y (Berndt-Evans-Williams, Theorem 2.5.1).

Example

a(1) = 1: q=7 = (0)^2+(0)*(1)+7*(1)^2, c = 2*(0)+1 = 1, ((1+3)*7-1)/27 = 1.
a(2) = -1: q=13 = (-3)^2+(-3)*(1)+7*(1)^2, c = 2*(-3)+1 = -5, ((-5+3)*13-1)/27 = -1.
a(3) = 7: q=19 = (4)^2+(4)*(-1)+7*(-1)^2, c = 2*(4)+(-1) = 7, ((7+3)*19-1)/27 = 7.

Prog

(PARI) e3(q) = for(b=-sqrtint(4*q\27)-1, sqrtint(4*q\27)+1, my(d=4*q-27*b^2, s); if(d>=0 && issquare(d, &s), for(j=0, 1, my(a=(-b+(-1)^j*s)/2); if(type(a)=="t_INT" && a^2+a*b+7*b^2==q && gcd(abs(a), abs(b))==1, my(c=2*a+b); if(c%3==1, return(((c+3)*q-1)/27)); if((-c)%3==1, return(((-c+3)*q-1)/27))))));
a(n) = my(cnt=0); forprime(q=7, +oo, if(q%3==1, cnt++; if(cnt==n, return(e3(q)))));

Crossrefs

Cf. A002476.

Sequence in context: A071117 A054221 A377933 * A195240 A253318 A277026

Adjacent sequences: A394564 A394565 A394566 * A394568 A394569 A394570

Keyword

sign

Author

Dao Van Tam, Mar 24 2026

Status

approved