A387776 Primes p = 1 (mod 8) for which the quartic Gauss sum is larger than the quadratic Gauss sum.

17, 73, 89, 113, 193, 257, 313, 337, 353, 401, 409, 449, 457, 521, 569, 577, 593, 601, 641, 673, 769, 857, 937, 977, 1049, 1097, 1129, 1153, 1193, 1201, 1217, 1249, 1289, 1321, 1361, 1409, 1481, 1489, 1601, 1609, 1697, 1721, 1801, 2129, 2281, 2297, 2377, 2393, 2417, 2521, 2593

Offset

1,1

Comments

For p = 1 (mod 2^k), the 2^(k-1)-adic, 2^(k-2)-adic, etc. Gauss sums g(d,p) are real. This comes from symmetry: both x and -x (mod p) are residues. This sequence is the set the primes p = 1 (mod 8) for which g(4,p) > g(2,p) =1 [if we divide each Gauss sum by sqrt(p)]. So for a prime, for which 2^d || (p-1), we can assign a sequence of + or - signs, the signs of (g(2^f,p) - g(2^(f-1),p)) f=2,3,...,(d-1)

Links

Table of n, a(n) for n=1..51.

Mathematica

d = 3; Select[Prime[Range[5000]], 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. A390354, A007519, A390758, A390759.

Sequence in context: A337704 A121243 A039371 * A245042 A144245 A307718

Adjacent sequences: A387773 A387774 A387775 * A387777 A387778 A387779

Keyword

nonn

Author

Zoltan Reti, Nov 23 2025

Status

approved