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A262587
"Special" prime powers in Serre's sense.
1
2, 3, 5, 7, 8, 13, 17, 31, 32, 37, 43, 73, 101, 128, 157, 197, 211, 241, 257, 307, 343, 401, 421, 463, 577, 601, 677, 757, 1123, 1297, 1483, 1601, 1723, 2048, 2187, 2551, 2917, 2971, 3137, 3307, 3541, 3907, 4357, 4423, 4831, 5113, 5477, 5701, 6007, 6163, 6481, 7057, 8011, 8101, 8191
OFFSET
1,1
COMMENTS
See Hirschfeld, pp. 49-50 for precise definition.
By a theorem of Hasse-Weil and Serre, every (absolutely irreducible, smooth) genus 2 curve over GF(q) has cardinality at most q + 1 + 2*floor(2*sqrt(q)). This sequence consists exactly of the prime powers q (excluding 4 and 9) for which there does not exist any genus 2 curve over GF(q) with cardinality equal to q + 1 + 2*floor(2*sqrt(q)). - Robin Visser, Aug 26 2023
REFERENCES
J. W. P. Hirschfeld, Linear codes and algebraic codes, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
J.-P. Serre, Oeuvres, vol. 3, pp. 658-663 and 664-669.
LINKS
Jean-Pierre Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.
Jean-Pierre Serre, Nombres de points des courbe algebriques sur F_q, Sémin. Théorie Nombres Bordeaux, 1982/83, No. 22; Oeuvres, vol. 3, pp. 664-669.
PROG
(Sage)
for q in range(1, 1000):
if Integer(q).is_prime_power():
p = Integer(q).prime_factors()[0]
if (not Integer(q).is_square()):
if ((floor(2*sqrt(q))%p == 0) or (q-1).is_square() or
(4*q-3).is_square() or (4*q-7).is_square()): print(q) # Robin Visser, Aug 26 2023
CROSSREFS
Subsequence of A246655.
Sequence in context: A028799 A080435 A108330 * A328724 A039892 A278645
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 21 2015
EXTENSIONS
More terms from Robin Visser, Aug 26 2023
STATUS
approved