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A290338
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Euler elliptic Carmichael numbers for the elliptic curve y^2 = x^3 + 80.
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2
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481, 1679, 1763, 3599, 4991, 5183, 6119, 7859, 9271, 9407, 9599, 18239, 24119, 24511, 24803, 31919, 38111, 38999, 46079, 56159, 57599, 58463, 62863, 63503, 67199, 72899, 82679, 152279, 163799, 167579, 170519, 181859, 187739, 196559, 208919, 213443, 236851
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OFFSET
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1,1
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COMMENTS
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An elliptic curve E over a field K is a nonsingular algebraic curve defined by a minimal Weierstrass equation
E/K: y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6
for some coefficients a_1, a_2, a_3, a_4, a_6 in K and discriminant of E not equal to zero.
Associated to E is an L-function L(E,s) = sum_{N} a_N / N^s. The map sending the positive integer N to a_N is a multiplicative function. Moreover, a_p = p + 1 - #E(F_p) with E(F_p) defined below, and a_{p^e} = a_p a_{p^{e-1}} - 1_E(p) p a_{p^{e-2}} for all e >= 2 where 1_E(p) is 1 if p does not divide the discriminant of E and is 0 otherwise.
Let E(Q) be an elliptic curve over Q, the field of rational numbers. We can replace y and x with scalar multiples such that the defining equation of E has integer coefficients.
For an integer N > 2, let E(Z/NZ) be the set of points (x,y) satisfying the defining equation of E in Z/NZ, the ring of integers modulo N, and the "points at infinity" (identity element). If the discriminant of E is coprime to N, then E(Z/NZ) forms an Abelian group.
If the discriminant of E is indivisible by a prime p and if #E(F_p) = p, then p is called an anomalous prime for E. There are no anomalous primes for the curve y^2 = x^3 + 80. This was shown in a paper by H. Qin (2016), see link.
The notions of strong elliptic pseudoprimes and strong elliptic Carmichael numbers are defined in "Anomalous Primes and Extension of the Korselt Criterion", see link. A Korselt criterion for these two notions was proven in "Anomalous Primes and Extension of the Korselt Criterion" based on the Korselt criterion developed in "Elliptic Carmichael Numbers and Elliptic Korselt Criteria" (J. Silverman, 2012).
Let N be a composite number, and P be a point of E(Z/NZ). Suppose that N+1-a_N is even, N has at least two distinct prime factors, and N is coprime to the discriminant of E. Then, N is an Euler elliptic pseudoprime for (E,P) if ((N+1-a_N)/2) P is the identity if P = 2Q for some Q in E(Z/NZ) and is of the form (x,y) where 2y + a_1 x + a_3 = 0 modulo N otherwise. For a prime p, let ord_p(N) be the p-adic order of N. Also let e_{N,p}(E) be the exponent of the group E(Z/p^(ord_p(N))Z). N is an Euler elliptic Carmichael number for E if and only if N has at least two distinct prime factors, N is coprime to the discriminant of E, and, for every prime p dividing N, 2e_{N,p}(E) divides N+1-a_N.
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LINKS
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I. Miyamoto and M. R. Murty, Elliptic Pseudoprimes, Mathematics of Computation, Vol. 53:187 (1989), 290-305.
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EXAMPLE
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Let N = 481 = 13*37. The discriminant of E: y^2 = x^3 + 80 is -16*(4*0^3 + 27*80^2) = -2764800, which is coprime to N. It turns out that E(Z/13Z) is isomorphic to the Abelian group Z/19Z and that E(Z/37Z) is isomorphic to the Abelian group Z/14Z + Z/2Z. In particular, #E(Z/13Z) = 19 and #E(Z/37Z) = 28, so a_13 = 13+1-19 = -5 and a_37 = 37+1-28 = 10. Therefore, a_N = a_13 * a_37 = -50, so N+1-a_N = 532. Moreover, e_{N,13} = 19 and e_{N,37} = 14, so 2*e_{N,13} = 38 and 2*e_{N,37} = 28 both divide N+1-a_N.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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