im interesting to know how it switches from algorithm trial division to another
until reached elliptic and quadratic seive .
How we can reached the implement of factorInteger[n]
Maybe this helps:
The implementation of the Multiple Polynomial Quadratic Sieve is based on code by Paul Zimmermann and Scott Contini, and it is described in the following articles.
Alford, W. R. and Pomerance, C. "Implementing the Self Initializing Quadratic Sieve on a Distributed Network. In Number Theoretic and Algebraic Methods in Computer Science, Proc. Internat. Moscow Conf., June-July 1993 (Ed. A. J. van der Poorten, I. Shparlinski, and H. G. Zimmer). Singapore, World Scientific, pp. 163-174, 1995.
Contini, S. "Factoring Integers with the Self-Initializing Quadratic Sieve", M.A. Thesis, University of Georgia, 1997.
Pomerance, C.; Smith, J. W.; and Tuler, R. "A Pipeline Architecture for Factoring Large Integers with the Quadratic Sieve Method." SIAM J. Comput. 17, pp. 387-403, 1988.
That note is, uhm, from Maplesoft documentation for the ifactor function in Maple.
What's notes give us Wolfram Research?