1—10 of 46 matching pages
Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.
Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.
§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 , where 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). …The theory of elliptic functions brings together complex analysis, algebraic curves, number theory, and geometry: Lang (1987), Siegel (1988), and Serre (1973).
§23.20(ii) Elliptic Curves►An algebraic curve that can be put either into the form …is an example of an elliptic curve (§22.18(iv)). … ►For extensive tables of elliptic curves see Cremona (1997, pp. 84–340). …
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 . Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard , 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.