A396871 - OEIS

%I #15 Jun 09 2026 15:15:57

%S 4,16,179,19,475,11875,17371,434275,375316,9382900,59879696,186796331,

%T 302588011,7564700275,25342996816,87659886836,826709586979,

%U 195925915636,4898147890900,122453697272500,502365711956491,4029220399392211,15431285989604579,65910059758111796

%N a(n) = 4*5^(2*n+1) - floor(2*5^(n+1/2))^2.

%C a(n) is the negated discriminants of the finite fields GF(3^(2*n+1)). Discriminants of finite fields defined in the thesis of Zaitsev and the Alekseenko et al paper to study algebraic curves over finite fields.

%C a(n) -> oo as n -> oo since each number can only occur finitely times: for each d, the equation x^2 + d = 4*5^n can be partitioned into three elliptic curves x^2 + d = 4*Y^3, x^2 + d = 4*5*Y^3, and x^2 + d = 4*5^2*Y^3, each having only finitely many integer solutions.

%H Jianing Song, <a href="/A396871/b396871.txt">Table of n, a(n) for n = 0..500</a>

%H E. Alekseenko, S. Aleshnikov, N. Markin and A. Zaytsev, <a href="https://arxiv.org/abs/0902.1901">Optimal Curves of Genus 3 over Finite Fields with Discriminant -19</a>, arXiv:0902.1901 [math.AG], 2009-2011. See Definition 1.1.

%H A.I. Zaitsev, <a href="https://hdl.handle.net/11245/1.279638">Optimality properties of curves over finite fields</a>, University of Armsterdam. See Section 3.1.

%t A396871[n_] := 4*5^(2*n + 1) - Floor[2*5^(n + 1/2)]^2;

%t Array[A396871, 25, 0] (* _Paolo Xausa_, Jun 09 2026 *)

%o (PARI) a(n,p=5) = my(r); sqrtint(4*p^(2*n+1),&r); r

%Y Cf. A396869.

%Y Cf. discriminants of finite fields whose order is an odd power of p: A201125 (p=2), A396870 (p=3), this sequence (p=5), A396872 (p=7).

%K nonn

%O 0,1

%A _Jianing Song_, Jun 08 2026