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1: 27.22 Software
  • Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below 10 16 . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard p 1 , and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

    For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

  • ECMNET Project. Links to software for elliptic curve methods of factorization and primality testing.

  • 2: 23.20 Mathematical Applications
    §23.20(ii) Elliptic Curves
    An algebraic curve that can be put either into the form …is an example of an elliptic curve22.18(iv)). … For extensive tables of elliptic curves see Cremona (1997, pp. 84–340). …
    3: 22.18 Mathematical Applications
    §22.18(i) Lengths and Parametrization of Plane Curves
    §22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem
    Algebraic curves of the form y 2 = P ( x ) , where P is a nonsingular polynomial of degree 3 or 4 (see McKean and Moll (1999, §1.10)), are elliptic curves, which are also considered in §23.20(ii). …For any two points ( x 1 , y 1 ) and ( x 2 , y 2 ) on this curve, their sum ( x 3 , y 3 ) , always a third point on the curve, is defined by the Jacobi–Abel addition law …
    4: 27.18 Methods of Computation: Primes
    The ECPP (Elliptic Curve Primality Proving) algorithm handles primes with over 20,000 digits. …
    5: 27.19 Methods of Computation: Factorization
    Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard p 1 algorithm, and the Elliptic Curve Method (ecm). …
    6: 19.30 Lengths of Plane Curves
    §19.30 Lengths of Plane Curves
    §19.30(i) Ellipse
    §19.30(ii) Hyperbola
    For other plane curves with arclength representable by an elliptic integral see Greenhill (1892, p. 190) and Bowman (1953, pp. 32–33).
    7: Bibliography M
  • Yu. I. Manin (1998) Sixth Painlevé Equation, Universal Elliptic Curve, and Mirror of 𝐏 2 . In Geometry of Differential Equations, A. Khovanskii, A. Varchenko, and V. Vassiliev (Eds.), Amer. Math. Soc. Transl. Ser. 2, Vol. 186, pp. 131–151.
  • H. McKean and V. Moll (1999) Elliptic Curves. Cambridge University Press, Cambridge.
  • 8: Bibliography E
  • ECMNET Project (website)
  • 9: 23.22 Methods of Computation
    Suppose that the invariants g 2 = c , g 3 = d , are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve23.20(ii)). …
    10: Bibliography K
  • N. Koblitz (1993) Introduction to Elliptic Curves and Modular Forms. 2nd edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York.