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%I #16 Dec 08 2025 15:03:02
%S 41,97,137,233,241,281,433,617,761,809,881,929,953,1009,1033,1297,
%T 1433,1553,1657,1753,1777,1873,1889,1913,1993,2017,2081,2089,2113,
%U 2137,2153,2161,2273,2441,2473,2617,2633,2713,2729,2753,2777,2801,2897,3041,3049,3137,3209,3761
%N 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).
%C 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.
%D Bruce C. Berndt, Ronald J. Evans and Kenneth S. Williams, Gauss and Jacobi Sums, Wiley Interscience, 1998.
%t d = 3;
%t Select[Prime[Range[1000]], Mod[#, 2^(d)] == 1 &&
%t Re[Sum[Exp[k^(2^(d - 1))*2*Pi*I/#] - Exp[k^(2^(d - 2))*2*Pi*I/#], {k, 0, # - 1}]/Sqrt[#]] < 0 &]
%Y Cf. A007519, A387776, A390758, A390759, A391135, A391136.
%K nonn
%O 1,1
%A _Zoltan Reti_, Dec 01 2025