A005523 a(n) = maximal number of rational points on an elliptic curve over GF(q), where q = A246655(n) is the n-th prime power > 1. (Formerly M3757)

5, 7, 9, 10, 13, 14, 16, 18, 21, 25, 26, 28, 33, 36, 38, 40, 43, 44, 50, 54, 57, 61, 64, 68, 75, 77, 81, 84, 88, 91, 97, 100, 102, 108, 117, 122, 124, 128, 130, 135, 144, 148, 150, 150, 154, 161, 163, 174, 176, 183, 189, 193, 196, 200, 206, 208, 219, 221, 226, 228, 241, 253, 258, 260

Offset

1,1

Comments

The successive values of q are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, ... (see A246655).

References

J. W. P. Hirschfeld, Linear codes and algebraic curves, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984. See N_q(1) on page 51.

J.-P. Serre, Oeuvres, vol. 3, pp. 658-663 and 664-669.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Robin Visser, Table of n, a(n) for n = 1..10000

Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.

W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

Eric Weisstein's World of Mathematics, Rational Point.

Wikipedia, Hasse's theorem on elliptic curves

Formula

a(n) <= q + 1 + 2*sqrt(q) where q = A246655(n) [Hasse theorem]. - Sean A. Irvine, Jun 26 2020

a(n) = q + 1 + floor(2*sqrt(q)) if p does not divide floor(2*sqrt(q)), q is a square, or q = p. Otherwise a(n) = q + floor(2*sqrt(q)) where q = A246655(n) [Waterhouse 1969]. - Robin Visser, Aug 02 2023

Example

a(1) = 5 because 5 is the maximal number of rational points on an elliptic curve over GF(2),
a(2) = 7 because 7 is the maximal number of rational points on an elliptic curve over GF(3),
a(3) = 9 because 9 is the maximal number of rational points on an elliptic curve over GF(4).

Prog

(Sage)
for q in range(1, 1000):
    if Integer(q).is_prime_power():
        p = Integer(q).prime_factors()[0]
        if (floor(2*sqrt(q))%p != 0) or (Integer(q).is_square()) or (q==p):
            print(q + 1 + floor(2*sqrt(q)))
        else:
            print(q + floor(2*sqrt(q)))  # Robin Visser, Aug 02 2023

Crossrefs

Cf. A000961, A246655.

Sequence in context: A184110 A138892 A190202 * A037084 A018935 A039501

Adjacent sequences: A005520 A005521 A005522 * A005524 A005525 A005526

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Reworded definition and changed offset so as to clarify the indexing. - N. J. A. Sloane, Jan 08 2017

More terms from Robin Visser, Aug 02 2023

Status

approved