A005524 Values k arising from a construction of Hirschfeld of k-arcs on elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1. (Formerly M0475)

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 18, 19, 20, 21, 22, 25, 27, 28, 30, 32, 34, 37, 38, 40, 42, 44, 45, 48, 50, 51, 54, 58, 61, 62, 64, 65, 67, 72, 74, 75, 75, 77, 80, 81, 87, 88, 91, 94, 96, 98, 100, 103, 104, 109, 110, 113, 114, 120, 126, 129, 130, 132, 135, 136, 137, 141

Offset

1,1

Comments

Let E be an elliptic curve over GF(q). A k-arc on E is a set of k points in E(GF(q)) such that no three are collinear (in the projective plane over GF(q)). Hirschfeld showed that if the number #E(GF(q)) of GF(q)-rational points on E is even, then there exists a k-arc on E for k = #E(GF(q))/2. Here, a(n) denotes the largest possible k arising from this construction, hence a(n) = floor(A005523(n)/2). Note that a(n) is not necessarily the maximal k such that there exists a k-arc on an elliptic curve over GF(q); e.g. the elliptic curve y^2 = x^3 + x + 1 over GF(5) contains a 6-arc consisting of the points {(0,1), (3,1), (4,2), (4,3), (0,4), (3,4)}. - Robin Visser, Aug 26 2023

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 M_q(1) on page 51.

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

J. W. P. Hirschfeld, G. Korchmáros, and F. Torres, Algebraic curves over a finite field, Princeton Ser. Appl. Math., Princeton University Press, Princeton, NJ, 2008.

Mathematica Information Center, Item 5175, for full code.

Ed Pegg Jr, Integer Complexity.

Formula

a(n) = floor(A005523(n)/2) [Hirschfeld]. - Robin Visser, Aug 26 2023

Example

For n = 4, the elliptic curve E : y^2 = x^3 + 3*x over GF(5) has 10 rational points.  As this is the maximal number of rational points an elliptic curve over GF(5) can have, this implies a(4) = 10/2 = 5. - Robin Visser, Aug 26 2023

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(floor((q + 1 + floor(2*sqrt(q)))/2))
        else:
            print(floor((q + floor(2*sqrt(q)))/2))  # Robin Visser, Aug 26 2023

Crossrefs

Cf. A000961 (values of q), A005523, A365216.

Sequence in context: A239348 A191881 A306424 * A191890 A247814 A082918

Adjacent sequences: A005521 A005522 A005523 * A005525 A005526 A005527

Keyword

nonn

Author

N. J. A. Sloane.

Extensions

New name and more terms from Robin Visser, Aug 26 2023

Status

approved