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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
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
EXTENSIONS
New name and more terms from Robin Visser, Aug 26 2023
STATUS
approved

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Last modified July 18 18:59 EDT 2024. Contains 374388 sequences. (Running on oeis4.)