A387776 - OEIS

%I #49 Dec 09 2025 16:46:10

%S 17,73,89,113,193,257,313,337,353,401,409,449,457,521,569,577,593,601,

%T 641,673,769,857,937,977,1049,1097,1129,1153,1193,1201,1217,1249,1289,

%U 1321,1361,1409,1481,1489,1601,1609,1697,1721,1801,2129,2281,2297,2377,2393,2417,2521,2593

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

%C 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)

%t 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 &]

%Y Cf. A390354, A007519, A390758, A390759.

%K nonn

%O 1,1

%A _Zoltan Reti_, Nov 23 2025