Number theory algorithms

# 2. Number theory algorithms

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.

##

2.1 Euclidean GCD algorithms

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.

##

2.2 Prime numbers: the Miller-Rabin test and its improvements

Small prime numbers $p<=65537$ 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.

###### 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].

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<n; instead we test only some x, so it may happen that we miss the right values of x that would expose the non-primality of n. So Fermat's test sometimes fails, i.e. says that n is a prime when n is in fact not a prime. Also there are infinitely many integers called "Carmichael numbers" which are not prime but pass the Fermat test for every base.

The Miller-Rabin algorithm improves on this by using the property that for prime n there are no nontrivial square roots of unity in the ring of integers modulo n (this is Lagrange's theorem). In other words, if x^2:=Mod(1,n) for some x, then x must be equal to 1 or -1 modulo n. (Since n-1 is equal to -1 modulo n, we have n-1 as a trivial square root of unity modulo n. Note that even if n is prime there may be nontrivial divisors of 1, for example, 2*49:=Mod(1,97).)

We can check that n is odd before applying any primality test. (A test n^2:=Mod(1,24) guarantees that n is not divisible by 2 or 3. For large n it is faster to first compute Mod(n,24) rather than n^2, or test n directly.) Then we note that in Fermat's test the number n-1 is certainly a composite number because n-1 is even. So if we first find the largest power of 2 in n-1 and decompose n-1=2^r*q with q odd, then x^(n-1):=Mod(a^2^r,n) where a:=Mod(x^q,n). (Here r>=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:

- Find the largest power of
2 in n-1 and an odd number q such that n-1=2^r*q.
- Select the "base number"
x<n. Compute the sequence a:=Mod(x^q,n), a^2, a^4, ..., a^2^r by repeated squaring modulo n. This sequence contains at least two elements since r>=1.
- 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 x<n if n is such that either 3*n+1 or 8*n+1 is a perfect square, or if n is a Carmichael number. Checking Carmichael numbers is slow, but it is easy to show that if n is a large enough prime number, then neither 3*n+1, nor 8*n+1, nor any s*n+1 with small integer s can be a perfect square. [If s*n+1=r^2, then s*n=(r-1)*(r+1).] Testing for a perfect square is quick and does not slow down the algorithm. This is however not implemented in Yacas because it seems that perfect squares are too rare for this improvement to be significant.

If an integer is not "strongly-probably-prime" for a given base b, then it is a composite number. However, the converse statement is false, i.e. "strongly-probably-prime" numbers can actually be composite. Composite strongly-probably-prime numbers for base b are called strong 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

###### 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`).

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<epsilon, it is enough to select k=1/Ln(4)*(Ln(n)-Ln(epsilon)).

Before calling `MillerRabin`, the function `IsPrime` performs two quick checks: first, for n>=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.

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.

##

2.3 Factorization of integers

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.)

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.

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<s[0]=Ln(n)/Ln(257). In other words, we only need to look for powers not greater than s[0]. This number can be approximated by the "integer logarithm" of n in base 257 (routine `IntLog(n, 257)`).

Now we need to check whether n is of the form p^s for s=2, 3, ..., s[0]. Note that if for example n=p^24 for some p, then the square root of n will already be an integer, n^(1/2)=p^12. Therefore it is enough to test whether n^(1/s) is an integer for all 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.

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]<n. It is clear that, on the average, a fraction 1/p of these integers will be divisible by p. Therefore (if t[k] are truly random) we should need on the average p tries until we find t[k] which is accidentally divisible by p. In practice, of course, we do not use a truly random sequence and the number of tries before we find a factor p may be significantly different from p. The quadratic polynomial seems to help reduce the number of tries in most cases.

But the Pollard "rho" algorithm may actually enter an infinite loop when the sequence x[k] repeats itself without giving any factors of n. For example, the unmodified "rho" algorithm starting from x[0]=2 loops on the number 703. The loop is detected by comparing x[2*k] and x[k]. When these two quantities become equal to each other for the first time, the loop may not yet have occurred so the value of GCD is set to 1 and the sequence is continued. But when the equality of x[2*k] and x[k] occurs many times, it indicates that the algorithm has entered a loop. A solution is to randomly choose a different starting number x[0] when a loop occurs and try factoring again, and keep trying new random starting numbers between 1 and n until a non-looping sequence is found. The current implementation stops after 100 restart attempts and prints an error message, "failed to factorize number".

A better (and faster) integer factoring algorithm needs to be implemented in Yacas.

Modern factoring algorithms are all probabilistic (i.e. they do not guarantee a particular finishing time) and fall into three categories:

- Methods that work well (i.e. quickly) if there is a relatively small factor
p of n (even if n itself is large). Pollard's "rho" algorithm belongs to this category. The fastest in this category is Lenstra's elliptic curves method (ECM).
- Methods that work equally quickly regardless of the size of factors (but slower with larger
n). These are the continued fractions method and the various "sieve" methods. The current best is the "General Number Field Sieve" (GNFS) but it is quite a complicated algorithm requiring operations with high-order algebraic numbers. The next best one is the "Multiple Polynomial Quadratic Sieve" (MPQS).
- Methods that are suitable only for numbers of special "interesting" form, e.g. Fermat numbers
2^2^k-1 or generally numbers of the form r^s+a where s is large but r and a are very small integers. The best method seems to be the "Special Number Field Sieve" which is a faster variant of the GNFS adapted to the problem.

There is ample literature describing these algorithms.

##

2.4 The Jacobi symbol

A number m is a "quadratic residue modulo n" if there exists a number k such that k^2:=Mod(m,n).

The Legendre symbol (m/ n) is defined as +1 if m is a quadratic residue modulo n and -1 if it is a non-residue. The Legendre symbol is equal to 0 if m/n is an integer.

The Jacobi symbol [m/n;] is defined as the product of the Legendre symbols of the prime factors f[i] of n=f[1]^p[1]*...*f[s]^p[s],

[m/n;]:=[m/f[1];]^p[1]*...*[m/f[s];]^p[s].
(Here we used the same notation [a/b;] for the Legendre and the Jacobi symbols; this is confusing but seems to be the current practice.) The Jacobi symbol is equal to 0 if m, n are not mutually prime (have a common factor). The Jacobi symbol and the Legendre symbol have values +1, -1 or 0.

The Jacobi symbol can be efficiently computed without knowing the full factorization of the number n. The currently used method is based on the following four identities for the Jacobi symbol:

[a/1;]=1.
[2/b;]=(-1)^((b^2-1)/8).
[(a*b)/c;]=[a/c;]*[b/c;].
- If
a:=Mod(b,c), then [a/c;]=[b/c;].
- If
a, b are both odd, then [a/b;]=[b/a;]*(-1)^((a-1)*(b-1)/4).

Using these identities, we can recursively reduce the computation of the Jacobi symbol [a/b;] to the computation of the Jacobi symbol for numbers that are on the average half as large. This is similar to the fast "binary" Euclidean algorithm for the computation of the GCD. The number of levels of recursion is logarithmic in the arguments a, b.

More formally, Jacobi symbol [a/b;] is computed by the following algorithm. (The number b must be an odd positive integer, otherwise the result is undefined.)

- If
b=1, return 1 and stop. If a=0, return 0 and stop. Otherwise, replace [a/b;] by [Mod(a,b)/b;] (identity 4).
- Find the largest power of
2 that divides a. Say, a=2^s*c where c is odd. Replace [a/b;] by [c/b;]*(-1)^(s*(b^2-1)/8) (identities 2 and 3).
- Now that
c<b, replace [c/b;] by [b/c;]*(-1)^((b-1)*(c-1)/4) (identity 5).
- Continue to step 1.

Note that the arguments a, b may be very large integers and we should avoid performing multiplications of these numbers. We can compute (-1)^((b-1)*(c-1)/4) without multiplications. This expression is equal to 1 if either b or c is equal to 1 mod 4; it is equal to -1 only if both b and c are equal to 3 mod 4. Also, (-1)^((b^2-1)/8) is equal to 1 if either b:=1 or b:=7 mod 8, and it is equal to -1 if b:=3 or b:=5 mod 8. Of course, if s is even, none of this needs to be computed.

##

2.5 Integer partitions

A partition of an integer n is a way of writing n as the sum of positive integers, where the order of these integers is unimportant. For example, there are 3 ways to write the number 3 in this way: 3=1+1+1, 3=1+2, 3=3. The function `PartitionsP` counts the number of such partitions.

###

Large n

The first algorithm used to compute this function uses the Rademacher-Hardy-Ramanujan (RHR) theorem and is efficient for large n. (See for example [Ahlgren 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))).

###

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.

##

2.6 Miscellaneous functions

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).

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.

##

2.7 Gaussian integers

A "Gaussian integer" is a complex number of the form z=a+b*I, where a and b are ordinary (rational) integers.

###### 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+i and its associates.
- The rational (ordinary) primes of the form
4*n+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.

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

- G. H. Hardy and E. M. Wright,
An Introduction to the Theory of Numbers. Oxford University Press (1945).
- H. Pollard,
The theory of Algebraic Numbers. Wiley, New York (1965).