A390758 Primes == 1 (mod 16), for which the octonic Gauss sum is greater than the quartic one.

17, 97, 113, 241, 257, 337, 401, 433, 641, 881, 977, 1009, 1153, 1249, 1409, 1553, 1697, 1777, 1873, 2017, 2081, 2113, 2129, 2273, 2593, 2609, 2689, 2833, 2897, 3089, 3217, 3329, 3361, 3457, 3697, 3761, 3793, 4177, 4241, 4273, 4289, 4481, 4513, 4657, 4673, 4801, 4817, 4993

Offset

1,1

Comments

If p==1 (mod 16) then both the quartic and the octonic Gauss sums are real numbers. The sign of the difference is undecided on page 164, Theorem 4.3.1 of "Gauss and Jacobi sums" by Berndt, Evans and Williams (1998).

References

Bruce C. Berndt, Ronald J. Evans and Kenneth S. Williams, Gauss and Jacobi Sums, Wiley Interscience, 1998, page 164.

Links

Table of n, a(n) for n=1..48.

Bruce C. Berndt and Ronald J. Evans, Sums of Gauss, Jacobi, and Jacobsthal, Journal of Number Theory, Volume 11, Issue 3, 1979, Pages 349-398.

Mathematica

Select[Prime[Range[100]], Mod[#, 2^(4)] == 1 &&
Re[Sum[Exp[k^(2^(3))*2*Pi*I/#] - Exp[k^(2^(2))*2*Pi*I/#], {k, 0, # - 1}]/Sqrt[#]] > 0 &]

Crossrefs

Cf. A094407, A155942, A390759, A387776, A391135, A391136.

Sequence in context: A264211 A094407 A131204 * A070186 A142189 A325068

Adjacent sequences: A390755 A390756 A390757 * A390759 A390760 A390761

Keyword

nonn

Author

Zoltan Reti, Nov 17 2025

Status

approved