This is a collection of files intended for use with PARI/GP:
http://pari.math.u-bordeaux.fr/

The main member of this sequence is auto.gp.c, which includes a number of other
.gp.c files.  When compiled, these files can be read into GP through the
install command.  The easiest way is to use gp2c:

gp2c-run auto.gp.c

which will compile the files then load them into a special session of gp. A
better solution is the script pari.sh which compiles them into a custom session.

The purpose of this collection is to provide new functions for gp.  Some are
designed to solve a problem which is not easily done in GP, while others merely
make a common task easier.

Ultimately, the goal for most of these functions is to incorporate them into
the mainline of PARI/GP, at which point they can be removed from this project.

== The Functions ==

The new commands:
Bell(n): Returns the n-th Bell or exponential number, Sloane's A000110.
bfile(name, v, offset=1): If v is given, creates a b-file with the values of v,
    using name as the A-number (given as a number or a filename). If v is not
    given, open the b-file name (again, as a filename or number) and return a
    vector of its values.
bigdiv(a,b,c,d): Does d divide a^b - c? Same as (a^b-c)%d == 0, but faster for
    large b. Example: bigdiv(2,p,1,d) checks if d divides the p-th Mersenne
    number.
bigfactor(a,b,c,lim,{start=2}): Find small prime factors of a^b - c (up to
    lim). Optional parameter start gives a starting point below which primes
    are not checked.
checkcmult(v,{verbose=1}): Is the sequence v completely multiplicative?
checkdiv(v,{verbose=1}): Is v a divisibility sequence?
checkmult(v,{verbose=1}): Is the sequence v multiplicative?
checkVDW(vv): Given a partition vv = [p1, p2, ...] with union(p1, p2, ...) =
    [1, 2, ..., n], finds a lower-bound proof for van der Waerden numbers based
    on the partition. Returns 0 if vv is not a partition of any initial
    segment, and k if vv proves that W(#vv, k) > n.
composite(n): Returns the n-th composite. Sloane's A002808.
contfracback(v, terms): Given a continued fraction v, gives the real number
    back. If terms is given, use only that many terms.
countPowerful(lim): Number of powerful numbers up to lim. Partial sum of
    characteristic function of of Sloane's A001694.
countSquarefree(lim): Counts the number of squarefree numbers up to lim.
deBruijnXi(x): Helper function for rhoest.  Finds a xi such that e^xi - 1 =
    x * xi.
DickmanRho(x): Estimates the value of the Dickman rho function. For x <= 3 the
    exact values are used, up to rounding; up to 15 the value is interpolated
    using known values and rhoest; after 15 rhoest is used, along with a
    correction factor based on the last value in rhoTable.
diffset(A, B): Set of all numbers of the form a-b, a in A, b in B.
Engel(x): Engel expansion of x.
Eng(n): English name of the number n.
eps(): Returns machine epsilon for the current precision.
Faulhaber(e,{a='x}): Returns the polynomial for the sum 1^e + 2^e + ... + x^e,
    evaluated at a.
fibmod(n,m): Returns the nth Fibonacci number mod m. Same as finonacci(n)%m,
    but faster for large n.
fnice(n): Returns a string with a 'nice' factorization of n.
fusc(n): Stern's diatomic series, which has many interpretations. Sloane's
    A002487.
gpf(n): The greatest prime factor of a number. Sloane's A006530.
isfactorial(n): Is n a factorial? Sloane's A012245; characteristic function of
    Sloane's A000142.
isFibonacci(n): Is n a Fibonacci number? Sloane's A010056; characteristic
    function of Sloane's A000045.
ispow2(n): Is n a power of two? Characteristic function of Sloane's A000079.
ispow3(n): Is n a power of three? Characteristic function of Sloane's A000244.
issemi(n): Is n a semiprime? Sloane's A064911; characteristic function of
    Sloane's A001358.
isthree(n): Is the number the sum of three squares? Sloane's A071374;
    characteristic function of Sloane's A000378.
istwo(n): Is the number a sum of two squares? Characteristic function of
    Sloane's A001481.
largestSquareFactor(n): Largest square dividing n. Sloane's A008833.
lg(x): Binary logarithm of x.
longestProgression1(v): Uses a quadratic algorithm of Jeff Erickson, which is
    worst-case optimal; better algorithms are available when there are long
    progressions (> lg #v lg lg #v).
longestProgression(v): Finds the longest arithmetic progression in v. Assumes
    that v is a vector of integers sorted from smallest to largest. Uses a
    space-efficient naive algorithm.
lpf(n): The least prime factor of a number. Sloane's A020639.
msb(n): Most significant bit of n: returns the greatest power of 2 <= the
    number. Sloane's A053644.
nice(o): Reformats the object o 'nicely' when possible. Currently chokes on
    multivariable polynomials.
normd(a,b): Amount of the normal distribution between a and b standard
    deviations. Plus/minus infinity coded as [+1]/[-1].
oddres(n): Returns the greatest odd number dividing n.
prp(n,b=2): Is n a b-probable prime?
rad(n): Radical of n, the largest squarefree number dividing n. Sloane's
    A007947.
rhoest(x): de Bruijn's asymptotic approximation for rho(x), rewritten as in
    van de Lune and Wattel 1969.  Curiously, their paper shows values for this
    estimate that differ from those calculated by this function, often as soon
    as the second decimal place -- but as the difference is in the direction of
    the true value, I have not looked further into this.
rnormal(): Returns a random normal variable with mean 0 and standard deviation
    1 at the current precision.
solvePell(n): Returns a solution to the equation x^2 - ny^2 = 1.
sopf(n): Sum of distinct prime factors of n. Sloane's A008472.
sopfr(n): Sum of prime factors of n (with multiplicity). Sloane's A001414.
sprp(n,b=2): Is n a b-strong probable prime?
sumset(A, B): Set of all numbers of the form a+b, a in A, b in B.
sumset_lim(A, B): Set of all numbers of the form a+b < lim, a in A, b in B.
toC(n): Format n for use with the PARI library (e.g., with gp2c programs).
ways2(n): Number of ways that n can be represented as a sum of two squares.
    Sloane's A000161.
ways3(n): Number of ways that n can be represented as a sum of three squares.
    Sloane's A000164.


Here are other functions that have implementations but which are not complete
or which have not been tested:
Mfactor(p,lim,{start=2}): Returns factors of the Mersenne number 2^p-1 up to lim, starting at start, provided p is a prime = 3 mod 4. Same as bigfactor(2,p,1,lim,start) but faster because it checks only factors of the form 2kp+1 that are +/- 1 mod 8.
tetrMod(a,b,M): Returns a^^b mod M.


Here are some internal functions which have been exposed to gp for testing
purposes.  Some of these may eventually be cleaned up for general use.
lnBell
issm3
consistency
cP
cS