A390354 Primes p == 1 (mod 8) for which the quartic Gauss sum (divided by sqrt(p)) is less than one, g(4,p) < 1 = g(2,p).

41, 97, 137, 233, 241, 281, 433, 617, 761, 809, 881, 929, 953, 1009, 1033, 1297, 1433, 1553, 1657, 1753, 1777, 1873, 1889, 1913, 1993, 2017, 2081, 2089, 2113, 2137, 2153, 2161, 2273, 2441, 2473, 2617, 2633, 2713, 2729, 2753, 2777, 2801, 2897, 3041, 3049, 3137, 3209, 3761

Offset

1,1

Comments

On page 160, in Theorem 4.2.1 of "Gauss & Jacobi Sums" by Berndt et al. the sign of the difference (g(4,p) - 1) is undecided.

References

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

Links

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

Mathematica

d = 3;
Select[Prime[Range[1000]], Mod[#, 2^(d)] == 1 &&
Re[Sum[Exp[k^(2^(d - 1))*2*Pi*I/#] - Exp[k^(2^(d - 2))*2*Pi*I/#], {k, 0, # - 1}]/Sqrt[#]] < 0 &]

Crossrefs

Cf. A007519, A387776, A390758, A390759, A391135, A391136.

Sequence in context: A039469 A124247 A256485 * A260556 A142658 A228573

Adjacent sequences: A390351 A390352 A390353 * A390355 A390356 A390357

Keyword

nonn

Author

Zoltan Reti, Dec 01 2025

Status

approved