Number theory algorithms
*INTRO This chapter describes the algorithms used for computing various number-theoretic functions.
We call "number-theoretic" any function that takes integer arguments,
produces integer values, and is of interest to number theory.
Euclidean GCD algorithms
*A GCD!binary Euclidean method
The main algorithm for the calculation of the GCD of two integers
is the binary Euclidean algorithm.
It is based on the following identities: $ Gcd(a,b) = Gcd(b,a) $,
$ Gcd(a,b) = Gcd(a-b,b)$, and for odd $b$, $Gcd(2*a,b) = Gcd(a,b)$.
Thus we can produce a sequence of pairs with the same GCD as the original two numbers, and each pair will be at most half the size of the previous pair.
The number of steps is logarithmic in the number of digits in $a$, $b$.
The only operations needed for this algorithm are binary shifts and subtractions (no modular division is necessary).
The low-level function for this is {MathGcd}.
To speed up the calculation when one of the numbers is much larger than another, one could use the property $Gcd(a,b)=Gcd(a,Mod(a,b))$.
This will introduce an additional modular division into the algorithm; this is a slow operation when the numbers are large.
Prime numbers: the Miller-Rabin test and its improvements
*A primality testing
Small prime numbers
*EVAL "$p<=" : (ToString()Write(FastIsPrime(0))) : "$"
are simply stored in a precomputed table as an array of bits; the bits corresponding to prime numbers are set to 1.
This makes primality testing on small numbers very quick.
This is implemented by the function {FastIsPrime}.
Primality of larger numbers is tested by the function {IsPrime} that uses the
Miller-Rabin algorithm.
*FOOT Initial implementation and documentation was supplied by Christian Obrecht.
This algorithm is deterministic (guaranteed correct within a certain running time) for
"small" numbers $n<3.4*10^13$ and probabilistic
(correct with high probability but not guaranteed) for larger numbers.
In other words, the Miller-Rabin test
could sometimes flag a large number $n$ as prime when in fact $n$ is
composite; but the probability for this to happen can be made extremely
small. The basic reference is [Rabin 1980].
We also implemented some of the improvements suggested in [Davenport 1992].
*A primality testing!Fermat test
*A primality testing!Miller-Rabin algorithm
The idea of the Miller-Rabin algorithm is to improve the Fermat primality test. If $n$ is prime, then for any $x$ we have $Gcd(n,x)=1$. Then by Fermat's "little theorem", $x^(n-1):=Mod(1,n)$. (This is really a simple statement; if $n$ is prime, then $n-1$ nonzero remainders modulo $n$: 1, 2, ..., $n-1$ form a cyclic multiplicative group.) Therefore we pick some "base" integer $x$ and compute $Mod(x^(n-1), n)$; this is a quick computation even if $n$ is large. If this value is not equal to $1$ for some base $x$, then $n$ is definitely not prime.
However, we cannot test *every* base $x=1$ since $n$ is odd.) In other words, the number $Mod(x^(n-1),n)$ is obtained by repeated squaring of the number $a$.
We get a sequence of $r$ repeated squares: $a$, $a^2$, ..., $a^(2^r)$.
The last element of this sequence must be $1$ if $n$ passes the Fermat test.
(If it does not pass, $n$ is definitely a composite number.)
If $n$ passes the Fermat test, the last-but-one element $a^(2^(r-1))$ of the sequence of squares is a square root of unity modulo $n$.
We can check whether this square root is non-trivial (i.e. not equal to $1$ or $-1$ modulo $n$). If it is non-trivial, then $n$ definitely cannot be a prime. If it is trivial and equal to $1$, we can check the preceding element, and so on. If an element is equal to $-1$, we cannot say anything, i.e. the test passes ($n$ is "probably a prime").
This procedure can be summarized like this:
* 1. Find the largest power of $2$ in $n-1$ and an odd number $q$ such that $n-1=2^r*q$.
* 2. Select the "base number" $x=1$.
* 3. If $a=1$ or $a=n-1$, the test passes on the base number $x$. Otherwise, the test passes if at least one of the elements of the sequence is equal to $n-1$ and fails if none of them are equal to $n-1$.
This simplified procedure works because the first element that is equal to $1$ *must* be preceded by a $-1$, or else we would find a nontrivial root of unity.
Here is a more formal definition.
An odd integer $n$ is called *strongly-probably-prime* for base $b$ if
$b^q:=Mod(1,n)$
or
$b^(q*2^i):=Mod(n-1,n)$ for some $i$ such that $0 <= i < r$,
where $q$ and $r$ are such that $q$ is odd and $n-1 = q*2^r$.
A practical application of this procedure needs to select particular base numbers.
It is advantageous (according to [Pomerance *et al.* 1980]) to choose *prime* numbers $b$ as bases, because for a composite base $b=p*q$, if $n$ is a strong pseudoprime for both $p$ and $q$, then it is very probable that $n$ is a strong pseudoprime also for $b$, so composite bases rarely give new information.
An additional check suggested by [Davenport 1992] is activated if $r>2$ (i.e. if $n:=Mod(1,8)$ which is true for only 1/4 of all odd numbers).
If $i>=1$ is found such that $b^(q*2^i):=Mod(n-1,n)$, then $b^(q*2^(i-1))$ is a square root of $-1$ modulo $n$.
If $n$ is prime, there may be only two different square roots of $-1$.
Therefore we should store the set of found values of roots of $-1$; if there are more than two such roots, then we woill find some roots $s1$, $s2$ of $-1$ such that $s1+s2!=Mod(0,n)$.
But $s1^2-s2^2:=Mod(0,n)$.
Therefore $n$ is definitely composite, e.g. $Gcd(s1+s2,n)>1$. This check costs very little computational effort but guards against some strong pseudoprimes.
Yet another small improvement comes from [Damgard *et al.* 1993].
They found that the strong primality test sometimes (rarely) passes on
composite numbers $n$ for more than $1/8$ of all bases $xstrong pseudoprimes for base $b$. There is a theorem that if $n$ is
composite, then among all numbers $b$ such that $1 < b < n$, at most one fourth
are such that $n$ is a strong pseudoprime for base $b$.
Therefore if $n$ is strongly-probably-prime for many bases, then the probability for $n$ to be composite is very small.
For numbers less than $B=34155071728321$, exhaustive
*FOOT And surely exhausting.
computations have shown
that there are no strong pseudoprimes simultaneously for bases 2, 3, 5, 7, 11, 13
and 17. This gives a simple and reliable primality test for integers below $B$.
If $n >= B$, the Rabin-Miller method consists in checking if $n$ is
strongly-probably-prime for $k$ base numbers $b$.
The base numbers are chosen to be consecutive "weak pseudoprimes" that are easy to generate (see below the function {NextPseudoPrime}).
*A primality testing!Miller-Rabin algorithm!choosing the bases
In the implemented routine {RabinMiller}, the number of bases $k$ is chosen to
make the probability of erroneously passing the test $p < 10^(-25)$. (Note that
this is *not* the same as the probability to give an incorrect answer,
because all numbers that do not pass the test are definitely composite.) The
probability for the test to pass mistakenly on a given number is found as
follows.
Suppose the number of bases $k$ is fixed. Then the probability for a
given composite number to pass the test is less than $p[f]=4^(-k)$. The
probability for a given number $n$ to be prime is roughly $p[p]=1/Ln(n)$ and to
be composite $p[c]=1-1/Ln(n)$. Prime numbers never fail the test.
Therefore, the probability for the test to pass is $p[f]*p[c]+p[p]$ and the probability to pass erroneously is
$$ p = (p[f]*p[c])/(p[f]*p[c]+p[p]) < Ln(n)*4^(-k) $$.
To make $p=4$ it checks that $n$ is not divisible by 2 or 3
(all primes larger than 4 must satisfy this);
second, for $n>257$, it checks that $n$ does not contain small prime factors $p<=257$.
This is checked by evaluating the GCD of $n$ with the precomputed product of all primes up to 257.
The computation of the GCD is quick and saves time in case a
small prime factor is present.
*A {NextPrime}
There is also a function {NextPrime(n)} that returns the smallest prime number larger than {n}.
This function uses a sequence 5,7,11,13,... generated by the function {NextPseudoPrime}.
This sequence contains numbers not divisible by 2 or 3 (but perhaps divisible by 5,7,...).
The function {NextPseudoPrime} is very fast because it does not perform a full primality test.
The function {NextPrime} however does check each of these pseudoprimes using {IsPrime} and finds the first prime number.
Factorization of integers
*A factorization of integers
*A {Factor}
When we find from the primality test that an integer $n$ is composite, we usually do not obtain any factors of $n$.
Factorization is implemented by functions {Factor} and {Factors}.
Both functions use the same algorithms to find all prime factors of a given integer $n$.
(Before doing this, the primality checking algorithm is used to detect whether $n$ is a prime number.)
Factorization consists of repeatedly finding a factor, i.e. an
integer $f$ such that $Mod(n, f)=0$, and dividing $n$ by $f$.
(Of course, each fastor $f$ needs to be factorized too.)
*A factorization of integers!small prime factors
First we determine whether the number $n$ contains "small" prime
factors $p<=257$. A quick test is to find the GCD of $n$ and the
product of all primes up to $257$: if the GCD is greater than 1, then
$n$ has at least one small prime factor. (The product of primes is
precomputed.) If this is the case, the trial division algorithm is
used: $n$ is divided by all prime numbers $p<=257$ until a factor is
found. {NextPseudoPrime} is used to generate the sequence of candidate
divisors $p$.
*A factorization of integers!checking for prime powers
After separating small prime factors, we test whether the number $n$ is an integer power of a prime number, i.e. whether $n=p^s$ for some prime number $p$ and an integer $s>=1$. This is tested by the following algorithm. We already know that $n$ is not prime and that $n$ does not contain any small prime factors up to 257. Therefore if $n=p^s$, then $p>257$ and $2<=s~~prime values of $s$ up to $s[0]$, and then we will definitely discover whether $n$ is a power of some other integer.
The testing is performed using the integer $n$-th root function {IntNthRoot} which quickly computes the integer part of $n$-th root of an integer number. If we discover that $n$ has an integer root $p$ of order $s$, we have to check that $p$ itself is a prime power (we use the same algorithm recursively). The number $n$ is a prime power if and only if $p$ is itself a prime power. If we find no integer roots of orders $s<=s[0]$, then $n$ is not a prime power.
*A factorization of integers!Pollard's "rho" algorithm
If the number $n$ is not a prime power, the Pollard "rho" algorithm is applied [Pollard 1978]. The Pollard "rho" algorithm takes an irreducible polynomial, e.g. $p(x)=x^2+1$ and builds a sequence of integers $x[k+1]:=Mod(p(x[k]),n)$, starting from $x[0]=2$. For each $k$, the value $x[2*k]-x[k]$ is attempted as possibly containing a common factor with $n$. The GCD of $x[2*k]-x[k]$ with $n$ is computed, and if $Gcd(x[2*k]-x[k],n)>1$, then that GCD value divides $n$.
The idea behind the "rho" algorithm is to generate an effectively
random sequence of trial numbers $t[k]$ that may have a common factor
with $n$. The efficiency of this algorithm is determined by the size of
the smallest factor $p$ of $n$. Suppose $p$ is the smallest prime
factor of $n$ and suppose we generate a random sequence of integers
$t[k]$ such that $1<=t[k]et al. 2001].)
The number of partitions $P(n)$ is equal to an infinite sum:
$$ P(n) = 1/(Pi*Sqrt(2))*Sum(k,1,Infinity, Sqrt(k)*A(k,n)*S(k,n)) $$,
where the functions $A$ and $S$ are defined as follows:
$$S(k,n) := Deriv(n) Sinh(Pi/k*Sqrt(2/3*(n-1/24)))/Sqrt(n-1/24)$$
$$ A(k,n) := Sum(l,1,k, delta(Gcd(l,k),1)*Exp(-2*Pi*I*(l*n)/k+Pi*I*B(k,l))) $$,
where $delta(x,y)$ is the Kronecker delta function (so that the summation goes only over integers $l$ which are mutually prime with $k$) and $B$ is defined by
$$ B(k,l) := Sum(j,1,k-1, j/k*(l*j/k-Floor(l*j/k)-1/2)) $$.
The first term of the series gives, at large $n$, the Hardy-Ramanujan asymptotic estimate,
$$ P(n) <> P_0(n) := 1/(4*n*Sqrt(3))*Exp(Pi*Sqrt((2*n)/3))$$.
The absolute value of each term decays quickly,
so after $O(Sqrt(n))$ terms the series gives an answer that is very close to the integer result.
There exist estimates of the error of this series, but they are complicated.
The series is sufficiently well-behaved and it is easier to determine the truncation point heuristically.
Each term of the series is either 0 (when all terms in $A(k,n)$ happen to cancel) or has a magnitude which is not very much larger than the magnitude of the previous nonzero term.
(But the series is not actually monotonic.)
In the current implementation, the series is truncated when $Abs(A(k,n)*S(n)*Sqrt(k))$ becomes smaller than $0.1$ for the first time;
in any case, the maximum number of calculated terms is $5+Sqrt(n)/2$.
One can show that asymptotically for large $n$, the required number of terms is less than $mu/Ln(mu)$, where $mu:=Pi*Sqrt((2*n)/3)$.
[Ahlgren ~~*et al.* 2001] mention that there exist explicit constants $B[1]$ and $B[2]$ such that
$$ Abs(P(n)-Sum(k,1,B[1]*Sqrt(n),A(k,n))) < B[2]*n^(-1/4)$$.
The floating-point precision necessary to obtain the integer result must be at least the number of digits in the first term $P_0(n)$, i.e.
$$ Prec > (Pi*Sqrt(2/3*n)-Ln(4*n*Sqrt(3)))/Ln(10) $$.
However, Yacas currently uses the fixed-point precision model.
Therefore, the current implementation divides the series by $P_0(n)$ and computes all terms to $Prec$ digits.
The RHR algorithm requires $O((n/Ln(n))^(3/2))$ operations, of which $O(n/Ln(n))$ are long multiplications at precision $Prec<>O(Sqrt(n))$ digits.
The computational cost is therefore $O(n/Ln(n)*M(Sqrt(n)))$.
*A partitions of an integer!by recurrence relation
Small $n$
The second, simpler algorithm involves a recurrence relation
$$ P[n] = Sum(k,1,n, (-1)^(k+1)*( P[n-k*(3*k-1)/2]+P[n-k*(3*k+1)/2] ) ) $$.
The sum can be written out as
$$ P(n-1)+P(n-2)-P(n-5)-P(n-7)+... $$,
where 1, 2, 5, 7, ... is the "generalized pentagonal sequence" generated by the pairs $k*(3*k-1)/2$, $k*(3*k+1)/2$ for $k=1$, 2, ...
The recurrence starts from $P(0)=1$, $P(1)=1$.
(This is implemented as {PartitionsP'recur}.)
The sum is actually not over all $k$ up to $n$ but is truncated when the pentagonal sequence grows above $n$.
Therefore, it contains only $O(Sqrt(n))$ terms.
However, computing $P(n)$ using the recurrence relation requires computing and storing $P(k)$ for all $1<=k<=n$.
No long multiplications are necessary, but the number of long additions of numbers with $Prec<>O(Sqrt(n))$ digits is $O(n^(3/2))$.
Therefore the computational cost is $O(n^2)$.
This is asymptotically slower than the RHR algorithm even if a slow $O(n^2)$ multiplication is used.
With internal Yacas math, the recurrence relation is faster for $n<300$ or so,
and for larger $n$ the RHR algorithm is faster.
Miscellaneous functions
*A divisors
The function {Divisors} currently returns the number of divisors of integer, while {DivisorsSum} returns the sum of these divisors.
(The current
algorithms need to factor the number.) The following theorem is used:
Let $ p[1]^k[1]* ... *p[r]^k[r] $ be the prime factorization of $n$, where $r$ is the
number of prime factors and $k[r]$ is the multiplicity of the $r$-th factor. Then
$$ Divisors(n) = (k[1]+1)*...*(k[r]+1) $$,
$$ DivisorsSum(n) = ( ((p[1]^(k[1]+1) - 1)/(p[1]-1)))*...*(p[r]^(k[r]+1)-1)/(p[r]-1) $$.
*A divisors!proper
The functions {ProperDivisors} and {ProperDivisorsSum} are functions that do the same as the above functions,
except they do not consider the number $n$ as a divisor for itself.
These functions are defined by:
$ ProperDivisors(n) = Divisors(n) - 1 $,
$ ProperDivisorsSum(n) = DivisorsSum(n) - n $.
Another number-theoretic function is {Moebius}, defined as
follows:
$Moebius(n)=(-1)^r$ if no factors of $n$ are repeated, $Moebius(n)=0$ if
some factors are repeated, and $Moebius(n)=1$ if $n = 1$.
This again requires to factor the number $n$ completely and investigate the
properties of its prime factors. From
the definition, it can be seen that if $n$ is prime, then $Moebius(n) = -1 $. The predicate
{IsSquareFree(n)} then reduces to $Moebius(n)!=0$, which means that no factors of $n$ are repeated.
Gaussian integers
*A Gaussian integers
A "Gaussian integer" is a complex number of the form $ z = a+b*I$, where $a$ and
$b$ are ordinary (rational) integers.
*FOOT To distinguish ordinary integers from Gaussian integers, the ordinary integers (with no imaginary part) are called "rational integers".
The ring of Gaussian integers is usually
denoted by $Z$[$I$] in the mathematical literature. It is an example of a ring of
algebraic integers.
The function {GaussianNorm} computes the norm $N(z)=a^2+b^2$ of $z$.
The norm
plays a fundamental role in the arithmetic of Gaussian integers, since it has
the multiplicative property:
$$ N(z.w) = N(z).N(w) $$.
A unit of a ring is an element that divides any other element of the ring.
There are four units in the Gaussian integers: $1$, $-1$, $I$, $-I$. They are
exactly the Gaussian integers whose norm is $1$. The predicate {IsGaussianUnit}
tests for a Gaussian unit.
Two Gaussian integers $z$ and $w$ are "associated" is $z/w$ is a unit.
For example, $2+I$ and $-1+2*I$ are associated.
A Gaussian integer is called prime if it is only divisible by the units and by
its associates. It can be shown that the primes in the ring of Gaussian
integers are:
* 1. $1+i$ and its associates.
* 2. The rational (ordinary) primes of the form $4*n+3$.
* 3. The factors $a+b*I$ of rational primes $p$ of the form $p=4*n+1$, whose norm is $p=a^2+b^2$.
For example, $7$ is prime as a Gaussian integer, while $5$ is not, since
$ 5 = (2+I)*(2-I) $.
Here $2+I$ is a Gaussian prime.
*A {Factors}
The ring of Gaussian integers is an example of an Euclidean ring, i.e. a ring
where there is a division algorithm.
This makes it possible to compute the greatest common divisor using Euclid's algorithm. This is what the function {GaussianGcd} computes.
As a consequence, one can prove a version of the fundamental
theorem of arithmetic for this ring: The expression of a Gaussian
integer as a product of primes is unique, apart from the order of primes, the
presence of units, and the ambiguities between associated primes.
The function {GaussianFactors} finds this expression of a Gaussian integer
$z$ as the product of Gaussian primes, and returns the result
as a list of pairs {{p,e}}, where $p$ is a Gaussian prime and $e$ is the corresponding exponent.
To do that, an auxiliary function called {GaussianFactorPrime} is used. This
function finds a factor of a rational prime of the form $4*n+1$. We
compute $a := (2*n)!$ (mod p). By Wilson's theorem $a^2$ is congruent to
$-1$ (mod $p$), and it follows that $p$ divides $(a+I)*(a-I)=a^2+1$ in the
Gaussian integers. The desired factor is then the {GaussianGcd} of $a+I$
and $p$. If the result is $a+b*I$, then $p=a^2+b^2$.
If $z$ is a rational (i.e. real) integer, we factor $z$ in the Gaussian integers by first
factoring it in the rational integers, and after that by factoring each of
the integer prime factors in the Gaussian integers.
If $z$ is not a rational integer, we find its possible Gaussian prime factors
by first factoring its norm $N(z)$ and then computing the exponent of each of
the factors of $N(z)$ in the decomposition of $z$.
References for Gaussian integers
* 1. G. H. Hardy and E. M. Wright,
*An Introduction to the Theory of Numbers*. Oxford University Press (1945).
* 2. H. Pollard,
*The theory of Algebraic Numbers*. Wiley, New York (1965).
A simple factorization algorithm for univariate polynomials
This section discusses factoring polynomials using
arithmetic modulo prime numbers. Information was
used from D. Knuth, *The Art of Computer Programming, Volume 2, Seminumerical Algorithms *
and J.H. Davenport et. al., *Computer Algebra, SYSTEMS AND ALGORITHMS FOR ALGEBRAIC COMPUTATION*.
A simple factorization algorithm is developed
for univariate polynomials. This algorithm is implemented
as the function {BinaryFactors}. The algorithm was named
the binary factoring algorithm since it determines
factors to a polynomial modulo $2^n$ for successive
values of $n$, effectively adding one binary digit to
the solution in each iteration. No reference to this
algorithm has been found so far in literature.
Berlekamp showed that polynomials can be efficiently factored
when arithmetic is done modulo a prime. The Berlekamp
algorithm is only efficient for small primes, but after that
Hensel lifting can be used to determine the factors modulo
larger numbers.
The algorithm presented here is similar in approach to applying
the Berlekamp algorithm to factor modulo a small prime, and then
factoring modulo powers of this prime (using the solutions found modulo
the small prime by the Berlekamp algorithm) by applying Hensel lifting.
However it is simpler in set up. It factors modulo 2, by trying
all possible factors modulo 2 (two possibilities, if the polynomial
is monic). This performs the same action usually left to the
Berlekamp step. After that, given a solution modulo $2^n$, it will
test for a solution $f_i$ modulo $2^n$ if $f_i$ or $f_i + 2^n$
are a solution modulo $2^(n+1)$.
This scheme raises the precision of the solution with one
digit in binary representation. This is similar to the linear
Hensel lifting algorithm, which factors modulo $p^n$ for some prime $p$, where
$n$ increases by one after each iteration. There is also a
quadratic version of Hensel lifting which factors modulo $p^2^n$,
in effect doubling the number of digits (in p-adic expansion) of the solution after
each iteration. However, according to "Davenport", the quadratic
algorithm is not necessarily faster.
The algorithm here thus should be equivalent in complexity
to Hensel lifting linear version. This has not been verified yet.
Modular arithmetic
This section copies some definitions and rules from
*The Art of Computer Programming, Volume 1, Fundamental Algorithms *
regarding arithmetic modulo an integer.
Arithmetic modulo an integer $p$ requires performing the
arithmetic operation and afterwards determining that
integer modulo $p$. A number $x$ can be written as
$$x=q*p+r$$
where $q$ is called the quotient, and $r$ remainder.
There is some liberty in the range one chooses $r$
to be in. If $r$ is an integer in the range ${0,1, ... ,(p-1)}$
then it is the *modulo*, $r = Mod(x,p)$.
When $Mod(x,p) = Mod(y,p)$, the notation $Mod(x=y,p)$
is used. All arithmetic calculations are done modulo
an integer $p$ in that case.
For calculations modulo some $p$ the following rules
hold:
* If $Mod(a=b,p)$ and $Mod(x=y,p)$, then
$Mod(a*x=b*y,p)$, $Mod(a+x=b+y,p)$, and $Mod(a-x=b-y,p)$.
This means that for instance also $Mod(x^n,p) = Mod(Mod(x,p)^n,p)$
* Two numbers $x$ and $y$ are *relatively prime* if they don't
share a common factor, that is, if their greatest common denominator
is one, $Gcd(x,y)=1$.
* If $Mod(a*x=b*y,p)$ and if $Mod(a=b,p)$, and if $a$ and
$p$ are relatively prime, then $Mod(x=y,p)$.
This is useful for dividing out common factors.
* $Mod(a=b,p)$ if and only if $Mod(a*n=b*n,n*p)$ when $n != 0$.
Also, if $r$ and $s$ are relatively prime, then $Mod(a=b,r*s)$ only if
$Mod(a=b,r)$ and $Mod(a=b,s)$.
These rules are useful when the modulus is changed.
For polynomials $v_1(x)$ and $v_2(x)$ it further holds
that
$$ Mod((v_1(x)+v_2(x))^p = v_1(x)^p + v_2(x)^p,p) $$
This follows by writing out the expression, noting that
the binomial coefficients that result are multiples of $p$,
and thus their value modulo $p$ is zero ($p$ divides these
coefficients), so only the two terms on the right hand side
remain.
Some corollaries
One corollary of the rules for calculations modulo an integer
is *Fermat's theorem, 1640* : if $p$ is a prime number
then
$$Mod(a^p=a,p)$$
for all integers $a$ (for a proof, see Knuth).
An interesting corollary to this is that, for some
prime integer $p$:
$$ Mod(v(x)^p = v(x^p),p) $$.
This follows from writing it out and using Fermat's theorem
to replace $a^p$ with $a$ where appropriate (the coefficients
to the polynomial when written out, on the left hand side).
Factoring using modular arithmetic
The task is to factor a polynomial
$$ p(x) = a_n*x^n + ... + a_0 $$
into a form
$$ p(x) = C*g(x)*f_1(x)^p_1*f_2(x)^p_2 * ... * f_m(x)^p_m $$
Where $f_i(x)$ are irreducible polynomials of the form:
$$ f_i(x) = x+c_i $$
The part that could not be factorized is returned as $g(x)$,
with a possible constant factor $C$.
The factors $f_i(x)$ and $g(x)$ are determined uniquely by requiring
them to be monic. The constant $C$ accounts for a common factor.
The $c_i$ constants in the resulting solutions $f_i(x)$ can be
rational numbers (or even complex numbers, if Gaussian integers
are used).
Preparing the polynomial for factorization
The final factoring algorithm needs the input polynomial to
be monic with integer coefficients (a polynomial is monic if
its leading coefficient is one). Given a non-monic
polynomial with rational coefficients, the following steps
are performed:
Convert polynomial with rational coefficients to polynomial with integer coefficients
First the least common multiple $lcm$ of the denominators of the
coefficients $p(x)$ has to be found, and the polynomial is multiplied by this number.
Afterwards, the $C$ constant in the result should have a factor
$1/lcm$.
The polynomial now only has integer coefficients.
Convert polynomial to a monic polynomial
The next step is to convert the polynomial to one where the leading
coefficient is one. In order to do so, following "Davenport",
the following steps have to be taken:
* 1. Multiply the polynomial by $a_n^(n-1)$
* 2. Perform the substitution $x=(y/a_n)$
The polynomial is now a monic polynomial in $y$.
After factoring, the irreducible factors of $p(x)$
can be obtained by multiplying $C$ with $1/(a_n^(n-1))$,
and replacing $y$ with $a_n*x$. The irreducible
solutions $a_n*x+c_i$ can be replaced by $x+c_i/a_i$
after multiplying $C$ by $a_n$, converting the factors
to monic factors.
After the steps described here the polynomial is now monic with integer coefficients,
and the factorization of this polynomial can be used to
determine the factors of the original polynomial $p(x)$.
Definition of division of polynomials
To factor a polynomial a division operation for polynomials
modulo some integer is needed. This algorithm needs to return
a quotient $q(x)$ and remainder $r(x)$ such that:
$$ Mod(p(x) = q(r)*d(x) + r(x),p) $$
for some polymomial $d(x)$ to be divided by, modulo
some integer p. $d(x)$ is
said to divide $p(x)$ (modulo $p$) if $r(x)$ is zero.
It is then a factor modulo $p$.
For binary factoring algorithm it is important
that if some monic $d(x)$ divides $p(x)$, then it also
divides $p(x)$ modulo some integer $p$.
Define $deg(f(x))$ to be the degree of $f(x)$ and $lc(f(x))$
to be the leading coefficient of $f(x)$. Then, if
$deg(p(x)) >= deg(d(x))$, one can compute an integer $s$
such that
$$ Mod(lc(d(x))*s = lc(p(x)),p) $$
If $p$ is prime, then
$$s = Mod(lc(p(x))*lc(d(x))^(p-2),p) $$
Because $Mod(a^(p-1) = 1,p)$ for any $a$. If $p$ is not prime
but $d(x)$ is monic (and thus $lc(d(x)) = 1$),
$$s = lc(p(x)) $$
This identity can also be used when dividing in
general (not modulo some integer), since the
divisor is monic.
The quotient can then be updated by adding
a term:
$term = s*x^(deg(p(x))-deg(d(x)))$
and updating the polynomial to be divided, $p(x)$,
by subtracting $d(x)*term$. The resulting polynomial
to be divided now has a degree one smaller than the
previous.
When the degree of $p(x)$ is less than the degree
of $d(x)$ it is returned as the remainder.
A full division algorithm for arbitrary integer $p>1$
with $lc(d(x)) = 1$ would thus look like:
divide(p(x),d(x),p)
q(x) = 0
r(x) = p(x)
while (deg(r(x)) >= deg(d(x)))
s = lc(r(x))
term = s*x^(deg(r(x))-deg(d(x)))
q(x) = q(x) + term
r(x) = r(x) - term*d(x) mod p
return {q(x),r(x)}
The reason we can get away with factoring modulo $2^n$
as opposed to factoring modulo some prime $p$ in later
sections is that the divisor $d(x)$ is monic. Its leading
coefficient is one and thus $q(x)$ and $r(x)$ can be
uniquely determined. If $p$ is not prime and $lc(d(x))$
is not equal to one, there might be multiple combinations
for which $p(x) = q(x)*d(x)+r(x)$, and we are interested
in the combinations where $r(x)$ is zero. This can be
costly to determine unless {q(x),r(x)} is unique.
This is the case here because we are factoring
a monic polynomial, and are thus only interested in cases
where $lc(d(x)) = 1$.
Determining possible factors modulo 2
We start with a polynomial $p(x)$ which is monic and
has integer coefficients.
It will be factored into a form:
$$ p(x) = g(x)*f_1(x)^p_1*f_2(x)^p_2 * ... * f_m(x)^p_m $$
where all factors $f_i(x)$ are monic also.
The algorithm starts by setting up a test polynomial,
$p_test(x)$ which divides $p(x)$, but has the property
that
$$ p_test(x) = g(x)*f_1(x)*f_2(x) * ... * f_m(x) $$
Such a polynomial is said to be *square-free*.
It has the same factors as the original polynomial, but
the original might have multiple of each factor,
where $p_test(x)$ does not.
The square-free part of a polynomial can be obtained
as follows:
$$ p_test(x) = p(x)/Gcd(p(x),D(x)p(x)) $$
It can be seen by simply writing this out that
$p(x)$ and $D(x)p(x)$ will have factors $f_i(x)^(p_i-1)$ in
common. these can thus be divided out.
It is not a requirement of the algorithm that the
algorithm being worked with is square-free, but it
speeds up computations to work with the square-free
part of the polynomial if the only thing sought after
is the set of factors. The multiplicity of the factors
can be determined using the original $p(x)$.
Binary factoring then proceeds by trying to find potential solutions
modulo $p=2$ first. There can only be two such
solutions: $x+0$ and $x+1$.
A list of possible solutions $L$ is set up with potential solutions.
Determining factors modulo $2^n$ given a factorization modulo 2
At this point there is a list $L$ with solutions modulo
$2^n$ for some $n$. The solutions will be of the form:
$x+a$. The first step is to determine if any of the
elements in $L$ divides $p(x)$ (not modulo any integer).
Since $x+a$ divides $p_test(x)$ modulo $2^n$,
both $x+a$ and $x+a-2^n$ have to be checked.
If an element in $L$ divides $p_test(x)$, $p_test(x)$
is divided by it, and a loop is entered to test how often
it divides $p(x)$ to determine the multiplicity $p_i$ of the
factor. The found factor $f_i(x) = x+c_i$ is added
as a combination ($x+c_i$, $p_i$). $p(x)$ is divided by $f_i(x)^p_i$.
At this point there is a list $L$ of factors that divide
$p_test(x)$ modulo $2^n$. This implies that for each
of the elements $u$ in $L$, either $u$ or $u+2^n$ should
divide $p_test(x)$ modulo $2^(n+1)$.
The following step is thus to set up a new list with new
elements that divide $p_test(x)$ modulo $2^(n+1)$.
The loop is re-entered, this time doing the calculation
modulo $2^(n+1)$ instead of modulo $2^n$.
The loop is terminated if the number of factors found
equals $deg(p_test(x))$, or if $2^n$ is larger than
the smallest non-zero coefficient of $p_test(x)$ as this
smallest non-zero coefficient is the product of all the
smallest non-zero coefficients of the factors, or if
the list of potential factors is zero.
The polynomial $p(x)$ can not be factored any further,
and is added as a factor ($p(x)$, $1$).
The function {BinaryFactors}, when implemented, yields
the following interaction in Yacas:
In> BinaryFactors((x+1)^4*(x-3)^2)
Out> {{x-3,2},{x+1,4}}
In> BinaryFactors((x-1/5)*(2*x+1/3))
Out> {{2,1},{x-1/5,1},{x+1/6,1}}
In> BinaryFactors((x-1123125)*(2*x+123233))
Out> {{2,1},{x-1123125,1},{x+123233/2,1}}
The binary factoring algorithm starts with a factorization modulo 2,
and then each time tries to guess the next bit of the
solution, maintaining a list of potential solutions.
This list can grow exponentially in certain instances.
For instance, factoring $(x-a)*(x-2*a)*(x-3*a)* ... $
implies a that the roots have common factors. There
are inputs where the number
of potential solutions (almost) doubles with each iteration.
For these inputs the algorithm becomes exponential. The worst-case
performance is therefore exponential. The list of potential
solutions while iterating will contain a lot of false roots
in that case.
Efficiently deciding if a polynomial divides another
Given the polynomial $p(x)$, and a potential divisor
$$ f_i(x) = x-p $$ modulo some $q=2^n$ an expression for
the remainder after division is
$$ rem(p)=Sum(i,0,n,a_i*p^i) $$
For the initial solutions modulo 2, where the possible
solutions are $x$ and $x-1$. For $p=0$, $rem(0) = a_0$.
For $p=1$, $rem(1) = Sum(i,0,n,a_i)$ .
Given a solution $x-p$ modulo $q=2^n$, we consider
the possible solutions $Mod(x-p,2^(n+1))$ and
$Mod(x-(p+2^n),2^n+1)$.
$x-p$ is a possible solution if $Mod(rem(p),2^(n+1)) = 0$.
$x-(p+q)$ is a possible solution if $Mod(rem(p+q),2^(n+1)) = 0$.
Expanding $Mod(rem(p+q),2*q)$ yields:
$$ Mod(rem(p+q),2*q) = Mod(rem(p) + extra(p,q),2*q) $$
When expanding this expression, some terms
grouped under $extra(p,q)$ have factors like $2*q$
or $q^2$. Since $q=2^n$, these terms vanish if the
calculation is done modulo $2^(n+1)$.
The expression for $extra(p,q)$ then becomes
$$ extra(p,q) = q*Sum(i,1,n/2, (2*i-1)*a(2*i)*p^(2*i-2)) $$
An efficient approach to determining if $x-p$ or $x-(p+q)$
divides $p(x)$ modulo $2^(n+1)$ is then to first calculate
$Mod(rem(p),2*q)$. If this is zero, $x-p$ divides $p(x)$.
In addition, if $Mod(rem(p)+extra(p,q),2*q)$ is zero,
$x-(p+q)$ is a potential candidate.
Other efficiencies are derived from the fact that the operations
are done in binary. Eg. if $q=2^n$, then $q_next=2^(n+1) = 2*q = q<<1 $ is
used in the next iteration. Also, calculations modulo $2^n$
are equivalent to performing a bitwise and with $2^n-1$. These
operations can in general be performed efficiently on
todays hardware which is based on binary representations.
Extending the algorithm
Only univariate polynomials with rational coefficients
have been considered so far. This could be extended to
allow for roots that are complex numbers $a+I*b$ where
both $a$ and $b$ are rational numbers.
For this to work the division algorithm would have to
be extended to handle complex numbers with integer
$a$ and $b$ modulo some integer, and the initial
setup of the potential solutions would have to be
extended to try $x+1+I$ and $x+I$ also. The step where
new potential solutions modulo $2^(n+1)$ are determined
should then also test for $x+I*2^n$ and $x+2^n+I*2^n$.
The same extension could be made for multivariate polynomials,
although setting up the initial irreducible polynomials
that divide $p_test(x)$ modulo 2 might become expensive
if done on a polynomial with many variables ($2^(2^m-1)$ trials
for $m$ variables).
Lastly, polynomials with real-valued coefficients *could*
be factored, if the coefficients were first converted to
rational numbers. However, for real-valued coefficients
there exist other methods (Sturm sequences).
Newton iteration
What the {BinaryFactor} algorithm effectively does is
finding a set of potential solutions modulo $2^(n+1)$ when
given a set of potential solutions modulo $2^n$.
There is a better algorithm that does something similar:
Hensel lifting. Hensel lifting is a generalized form of
Newton iteration, where given a factorization modulo
$p$, each iteration returns a factorization modulo $p^2$.
Newton iteration is based on the following idea: when one
takes a Taylor series expansion of a function:
$$f(x[0]+dx) := f(x[0]) + (D(x)f(x[0]))*dx + ... $$
Newton iteration then proceeds by taking only the first
two terms in this series, the constant plus the constant
times $dx$. Given some good initial value $x_0$, the function
will is assumed to be close to a root, and the function
is assumed to be almost linear, hence this approximation.
Under these assumptions, if we want $f(x_0+dx)$ to be zero,
$$f(x[0]+dx) = f(x[0]) + (D(x)f(x[0]))*dx = 0 $$
This yields:
$$ dx := -f(x[0])/(D(x)f(x[0])) = 0 $$
And thus a next, better, approximation for the root is
$x[1]:=x_0-f(x[0])/(D(x)f(x[0]))$, or more general:
$$ x[n+1] =x[n]-f(x[n])/(D(x)f(x[n]))$$
If the root has multiplicity one, a Newton iteration can
converge *quadratically*, meaning the number of
decimals precision for each iteration doubles.
As an example, we can try to find a root of $Sin(x)$ near
$3$, which should converge to $Pi$.
Setting precision to 30 digits,
In> Builtin'Precision'Set(30)
Out> True;
We first set up a function $dx(x)$:
In> dx(x):=Eval(-Sin(x)/(D(x)Sin(x)))
Out> True;
And we start with a good initial approximation to $Pi$,
namely $3$. Note we should set {x} *after* we set
dx(x), as the right hand side of the function definition is
evaluated. We could also have used a different parameter name
for the definition of the function $dx(x)$.
In> x:=3
Out> 3;
We can now start the iteration:
In> x:=N(x+dx(x))
Out> 3.142546543074277805295635410534;
In> x:=N(x+dx(x))
Out> 3.14159265330047681544988577172;
In> x:=N(x+dx(x))
Out> 3.141592653589793238462643383287;
In> x:=N(x+dx(x))
Out> 3.14159265358979323846264338328;
In> x:=N(x+dx(x))
Out> 3.14159265358979323846264338328;
As shown, in this example the iteration converges quite
quickly.
Finding roots of multiple equations in multiple variables using Newton iteration
One generalization, mentioned in W.H. Press et al.,
*NUMERICAL RECIPES in C, The Art of Scientific computing*
is finding roots for multiple functions in multiple variables.
Given $N$ functions in $N$ variables, we want to solve
$$ f_i(x[1],...,x[N]) = 0 $$
for $i = 1 .. N $. If de denote by $X$ the vector
$$ X := {x[1],x[2],...,x[N]} $$
and by $dX$ the delta vector, then one can write
$$ f_i(X+dX) := f_i(X)+Sum(j,1,N,(D(x_j)f_i(X)))*dx[j] $$
Setting $f_i(X+dX)$ to zero, one obtains
$$ Sum(j,1,N,a[i][j]*dx_j)=b[i]$$
where
$$a[i][j] := D(x_j)f_i(X)$$
and
$$b_i := -f_i(X)$$
So the generalization is to first initialize $X$ to a good
initial value, calculate the matrix elements $a[i][j]$ and
the vector $b[i]$, and then to proceed to calculate $dX$
by solving the matrix equation, and calculating
$$X[i+1] := X[i] + dX[i]$$
In the case of one function with one variable, the summation
reduces to one term, so this linear set of equations was
a lot simpler in that case. In this case we will have to solve
this set of linear equations in each iteration.
As an example, suppose we want to find the zeroes for the
following two functions:
$$f_1(a,x) := Sin(a*x)$$
and
$$f_2(a,x) := a-2$$
It is clear that the solution to this is $a=2$ and $x:=N*Pi/2$
for any integer value $N$.
We will do calculations with precision 30:
In> Builtin'Precision'Set(30)
Out> True;
And set up a vector of functions ${f_1(X),f_2(X)}$
where $X:={a,x}$
In> f(a,x):={Sin(a*x),a-2}
Out> True;
Now we set up a function {matrix(a,x)} which returns the
matrix $a[i][j]$:
In> matrix(a,x):=Eval({D(a)f(a,x),D(x)f(a,x)})
Out> True;
We now set up some initial values:
In> {a,x}:={1.5,1.5}
Out> {1.5,1.5};
The iteration converges a lot slower for this example, so we
will loop 100 times:
In> For(ii:=1,ii<100,ii++)[{a,x}:={a,x}+\
N(SolveMatrix(matrix(a,x),-f(a,x)));]
Out> True;
In> {a,x}
Out> {2.,0.059667311457823162437151576236};
The value for $a$ has already been found. Iterating a
few more times:
In> For(ii:=1,ii<100,ii++)[{a,x}:={a,x}+\
N(SolveMatrix(matrix(a,x),-f(a,x)));]
Out> True;
In> {a,x}
Out> {2.,-0.042792753588155918852832259721};
In> For(ii:=1,ii<100,ii++)[{a,x}:={a,x}+\
N(SolveMatrix(matrix(a,x),-f(a,x)));]
Out> True;
In> {a,x}
Out> {2.,0.035119151349413516969586788023};
the value for $x$ converges a lot slower this time, and to the uninteresting
value of zero (a rather trivial zero of this set of functions).
In fact for all integer values $N$ the value $N*Pi/2$ is a solution.
Trying various initial values will find them.
Newton iteration on polynomials
von zur Gathen et al., *Modern Computer algebra* discusses
taking the inverse of a polynomial using Newton iteration.
The task is, given a polynomial $f(x)$, to find a polynomial
$g(x)$ such that $f(x) = 1/g(x)$, modulo some power in x.
This implies that we want to find a polynom $g$ for which:
$$h(g) = 1/g-f = 0$$
Applying a Newton iteration step $g[i+1] = g[i] - h(g[i])/(D(g)h(g[i]))$
to this expression yields:
$$g[i+1] = 2*g[i] - f*(g[i])^2$$
von zur Gathen then proves by induction that for $f(x)$ monic,
and thus $f(0)=1$, given initial value $g_0(x) = 1$, that
$$Mod(f*g_i=1,x^(2^i))$$
Example:
suppose we want to find the polynomial $g(x)$ up to the 7th degree
for which $Mod(f(x)*g(x) = 1,x^8)$, for the function
$$ f(x):=1+x+x^2/2+x^3/6+x^4/24 $$
First we define the function f:
In> f:=1+x+x^2/2+x^3/6+x^4/24
Out> x+x^2/2+x^3/6+x^4/24+1;
And initialize $g$ and $i$.
In> g:=1
Out> 1;
In> i:=0
Out> 0;
Now we iterate, increasing $i$, and replacing $g$ with the
new value for $g$:
In> [i++;g:=BigOh(2*g-f*g^2,x,2^i);]
Out> 1-x;
In> [i++;g:=BigOh(2*g-f*g^2,x,2^i);]
Out> x^2/2-x^3/6-x+1;
In> [i++;g:=BigOh(2*g-f*g^2,x,2^i);]
Out> x^7/72-x^6/72+x^4/24-x^3/6+x^2/2-x+1;
The resulting expression must thus be:
$$g(x):=x^7/72-x^6/72+x^4/24-x^3/6+x^2/2-x+1$$
We can easily verify this:
In> Expand(f*g)
Out> x^11/1728+x^10/576+x^9/216+(5*x^8)/576+1;
This expression is 1 modulo $x^8$, as can easily be shown:
In> BigOh(%,x,8)
Out> 1;