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 A199263 Let CardE(K) the number of elements (x, y) of the finite group defined by the elliptic curve y^2 = x^3 + x + 1 (mod p) including the point at infinity ; a(n) is the difference between 2*sqrt(p) and  |cardE(K) - (p + 1)|. 2
 2, 3, 1, 2, 4, 3, 8, 7, 5, 4, 10, 2, 5, 3, 1, 10, 12, 3, 4, 3, 15, 11, 12, 8, 18, 17, 3, 17, 7, 10, 20, 18, 11, 9, 10, 22, 12, 0, 1, 24, 26, 18, 2, 20, 4, 10, 18, 9, 30, 28, 27, 8, 9, 1, 23, 28, 8, 30, 11, 26, 13, 8, 28, 32, 3, 20, 26, 22, 21, 23, 33, 28, 16 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Hasse’s theorem tells us the following estimate : |cardE(K) - (p + 1)|<= 2*sqrt(p) where K = Fp is the finite field with p elements and E an elliptic curve y^2 = x^3 + x + 1 (mod p) defined over K. CardE(K) is the number of elements (x, y) of the finite group defined by the elliptic curve  including the point at infinity. The number of points of the curve grows roughly as the number of elements in the field. This sequence gives the integer difference  : 2*sqrt(p) - |cardE(K) - (p + 1)|. We obtain remarkable values such that a(38) = 0, a(258) = 0. LINKS Michel Lagneau, Table of n, a(n) for n = 1..1000 Wolfram MathWorld, , Elliptic Curve Group Law. Wikipedia, Hasse's theorem on elliptic curves EXAMPLE For n=6, p = prime(6)= 13 and a(6) = 3  because the  solutions of y^2 = x^3 + x + 1 (mod 13) are {(inf, inf), (7,0), (0,1), (5, 1), (8,1), (4,2), (11,2), (1, 4), (12,5), (10,6), (10,7), (12, 8), (1,9), (4,11), (11,11), (0, 12), (5,12), (8,12)} => CardE(K) = A192334(6) = 18, and floor(2*sqrt(13) - 18 + 13 + 1) = floor(7.2111025 -4) = 3. MAPLE for m from 1 to 100 do:p:=ithprime(m):it:=1:for y from 0 to p-1 do:for x from 0 to p-1 do:z:=x^3+x+1:z1:=irem(z, p):z2:=irem(y^2, p):if z1=z2 then it:=it+1:else fi:od:od:y:=  floor(2*evalf(sqrt(p))-abs(it-p-1)): printf(`%d, `, y):od: CROSSREFS Cf. A192334, A098514. Sequence in context: A035459 A048232 A163256 * A257669 A181803 A144962 Adjacent sequences:  A199260 A199261 A199262 * A199264 A199265 A199266 KEYWORD nonn AUTHOR Michel Lagneau, Nov 07 2011 STATUS approved

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