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
Eric Weisstein's World of Mathematics, 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
KEYWORD
nonn
AUTHOR
Michel Lagneau, Nov 07 2011
STATUS
approved