A005523 - OEIS (original) (raw)

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)

9

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

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).

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

EXTENSIONS

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