Function: factorint
Section: number_theoretical
C-Name: factorint
Prototype: GD0,L,
Help: factorint(x,{flag=0}): factor the integer x. flag is optional, whose
 binary digits mean 1: avoid MPQS, 2: avoid first-stage ECM (may fall back on
 it later), 4: avoid Pollard-Brent Rho and Shanks SQUFOF, 8: skip final ECM
 (huge composites will be declared prime).
Doc: factors the integer $n$ into a product of
 pseudoprimes (see \kbd{ispseudoprime}), using a combination of the
 \idx{Shanks SQUFOF} and \idx{Pollard Rho} method (with modifications due to
 Brent), \idx{Lenstra}'s \idx{ECM} (with modifications by Montgomery), and
 \idx{MPQS} (the latter adapted from the \idx{LiDIA} code with the kind
 permission of the LiDIA maintainers), as well as a search for pure powers.
 The output is a two-column matrix as for \kbd{factor}: the first column
 contains the ``prime'' divisors of $n$, the second one contains the
 (positive) exponents.

 By convention $0$ is factored as $0^{1}$, and $1$ as the empty factorization;
 also the divisors are by default not proven primes if they are larger than
 $2^{64}$, they only failed the BPSW compositeness test (see
 \tet{ispseudoprime}). Use \kbd{isprime} on the result if you want to
 guarantee primality or set the \tet{factor_proven} default to $1$.
 Entries of the private prime tables (see \tet{addprimes}) are also included
 as is.

 This gives direct access to the integer factoring engine called by most
 arithmetical functions. \fl\ is optional; its binary digits mean 1: avoid
 MPQS, 2: skip first stage ECM (we may still fall back to it later), 4: avoid
 Rho and SQUFOF, 8: don't run final ECM (as a result, a huge composite may be
 declared to be prime). Note that a (strong) probabilistic primality test is
 used; thus composites might not be detected, although no example is known.

 You are invited to play with the flag settings and watch the internals at
 work by using \kbd{gp}'s \tet{debug} default parameter (level 3 shows
 just the outline, 4 turns on time keeping, 5 and above show an increasing
 amount of internal details).
