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A317174
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Elliptic Carmichael numbers for the elliptic curve y^2 = x^3 + 80.
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0
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481, 629, 703, 1679, 1763, 1769, 3599, 4991, 5183, 6119, 6989, 7783, 7859, 8797, 8987, 9271, 9407, 9599, 12209, 13817, 14219, 18239, 20999, 24119, 24511, 24803, 26333, 31919, 36577, 38111, 38999, 44099, 46079, 56159, 57599, 58463, 62863, 63503, 67199, 67889, 68741, 70859, 71819, 72899, 76751
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OFFSET
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1,1
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COMMENTS
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Let p>3 be a prime and Z/pZ the field of integers modulo p. An elliptic curve E over Z/pZ, denoted by E(Z/pZ), is a set of points (x,y) in Z/pZ x Z/pZ such that y^2 = x^3 + ax + b with discriminant not equal to zero (4*a^3 + 27*b^2 != 0), and an additional point O, called the "point at infinity".
An elliptic curve can be seen as an additive Abelian group with the point at infinity as an identity element. The order of the elliptic curve, the number of points including the point at infinity, is denoted by #E(Z/pZ). There is another equivalent definition of elliptic curve in projective coordinates. Namely, the elliptic curve E(Z/pZ) is a set of points (x:y:z) in P^2(Z/pZ) that satisfy the equation y^2z = x^3 + axz^2 + bz^3. Here, the points (x,y) are mapped to (x:y:1), and O is mapped to (0:1:0). The formulas for computing multiples and adding points can be found in "Elliptic Curves: Number Theory and Cryptography" by L. C. Washington.
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 "point at infinity" (identity element). If the discriminant of E is coprime to N, then E(Z/NZ) forms an Abelian group.
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(Z/pZ) with E(Z/pZ) 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.
The notions of elliptic pseudoprimes and elliptic Carmichael numbers are defined in "Elliptic Carmichael Numbers and Elliptic Korselt Criteria", see link. Let N be a composite number, and P be a point of E(Z/NZ). Suppose that N has at least two distinct prime factors and N is coprime to the discriminant of E. Then, N is an elliptic pseudoprime for (E,P) if (N+1-a_N)P is the identity. N is a Carmichael number for E if it is a pseudoprime at (E,P) for all point P on E.
A Korselt criterion for the notions of elliptic pseudoprimes and elliptic Carmichael numbers was proved in "Elliptic Carmichael Numbers and Elliptic Korselt Criteria", see link. 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). Then N is an 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, e_{N,p}(E) divides N+1-a_N.
The resulting sequence is based on work done during the REU program, "Complexity Across Disciplines", supported by the National Science Foundation under the grant DMS -1659872.
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REFERENCES
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L. C. Washington, Elliptic Curves: Number Theory and Cryptography, Champan and Hall, (2008).
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LINKS
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EXAMPLE
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Let N = 481=13*37. The discriminant of E is -16*(4*0^3 + 27*80^2) = -2764800, which is coprime to N. It can be computed that E(Z/13Z) = Z/19Z, and so a_13 = 13 + 1 - 19 = -5 and e_{N,13}(E) = 19. Similarly, E(Z/37Z) = Z/2Z+Z/14Z, so a_37 = 37 + 1 - 28 = 10 and e_{N,37}(E) = 14. Then a_481 = -50, so N+1-a_N = 481 + 1 + 50 = 532, which is divisible by both e_{N,13}(E) and e_{N,37}(E). Hence N is an elliptic Carmichael number for E.
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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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