OFFSET
1,1
COMMENTS
a(5) through a(14) are identical to the first 10 values of q, the left-hand column of "Example 2.3. We give examples of maximal and minimal elliptic curves over finite fields over F_q with discriminant -19 for all q < 1000", p. 4, and "Example 5.2. We produce examples of optimal curves over finite fields with discriminant -19" pp. 10-11 of E. Alekseenko, et al. - Jonathan Vos Post, Feb 12 2009
The discriminant of F_q defined in the E. Alekseenko et al paper in floor(2*sqrt(q))^2 - 4*q (A396869). Note that the only solution to x^2 + 19 = 4*p^n, p prime, n >= 3 is (x,p,n) = (559,5,7): the case p != 19 is treated in Theorem 1.1 of the Luca et al link, and the equation for p = 19 can be partitioned into three elliptic curves X^2 + X + 5 = Y^3, X^2 + X + 5 = 19*Y^3, and X^2 + X + 5 = 19^2*Y^3, each having no integer solutions. As a result, F_q has discriminant -19 if and only if q >= 47 is in this sequence or q = 78125. - Jianing Song, Jun 09 2026
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
E. Alekseenko, S. Aleshnikov, N. Markin and A. Zaytsev, Optimal Curves of Genus 3 over Finite Fields with Discriminant -19, arXiv:0902.1901 [math.AG], 2009-2011.
Patrick De Geest, World!Of Numbers
Florian Luca, Szabolcs Tengely, and Alain Togbé, On the Diophantine equation x^2 + C = 4y^n.
FORMULA
a(n) >> n^2 log n (Brun sieve). - Charles R Greathouse IV, Nov 01 2022
MATHEMATICA
nn = Range[0, 200]; Select[nn^2 + nn + 5, PrimeQ] (* Jean-François Alcover, Nov 17 2018 *)
PROG
(Magma) [a: n in [0..250]|IsPrime(a) where a is n^2+n+5]; // Vincenzo Librandi, Dec 20 2010
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved