The functions in the NumTheory file are predominantly basic
operations from number theory. Most of the functions may be
applied to machine integers or big integers (i.e. values of
type BigInt). Please recall that computational number theory is
not the primary remit of CoCoALib, so do not expect to find a
complete collection of operations here -- you would do better to
look at Victor Shoup's NTL (Number Theory Library), or PARI/GP,
or some other specialized library/system.
See also BigIntOps for very basic arithmetic operations on integers,
and BigRat for very basic arithmetic operations on rational numbers.
Several of these functions give errors if they are handed unsuitable values:
unless otherwise indicated below the error is of type ERR::BadArg.
All functions expecting a modulus will throw an error if the modulus is
less than 2 (or an unsigned long value too large to fit into a long).
The main functions available are:
gcd(m,n) computes the non-negative gcd of m and n. If both args are machine integers, the result is of type long (or error if it does not fit); otherwise result is of type BigInt.
IsCoprime(m,n) returns true iff abs(gcd(m,n)) == 1
ExtGcd(a,b,m,n) computes the non-negative gcd of m and n; also sets a and b so that gcd = a*m+b*n. If m and n are machine integers then a and b must be of type (signed) long. If m and n are of type BigInt then a and b must also be of type BigInt. The cofactors a and b satisfy abs(a) <= abs(n)/2g and abs(b) <= abs(m)/2g where g is the gcd (inequalities are strict if possible). Error if m=n=0.
InvMod(r,m) computes the least positive inverse of r modulo m; throws error (ERR::DivByZero) if the inverse does not exist. Throws error (ERR::BadModulus) if m < 2 (or too big for long). Result is of type long if m is a machine integer; otherwise result is of type BigInt.
InvMod(r,m, RtnZeroOnError) same as InvMod(r,m) except that it returns 0 if the inverse does not exist; RtnZeroOnError comes from an enum.
InvModNoArgCheck(r,m) computes the least positive inverse of r modulo m; ASSUMES 0 <= r < m and 2 <= m <= MaxLong; result is a long. Throws error ERR::DivByZero if gcd(r,m) is not 1.
lcm(m,n) computes the non-negative lcm of m and n. If both args are machine integers, the result is of type long; otherwise result is of type BigInt. Gives error ERR::ArgTooBig if the lcm of two machine integers is too large to fit into a long.
There is a class called CoprimeFactorBasis_BigInt for computing a coprime
factor basis of a set of integers:
CoprimeFactorBasis_BigInt() default ctor; base is initially empty.
CFB.myAddInfo(n) use also the integer n when computing the factor base.
CFB.myAddInfo(v) use also the elements of std::vector<long> v or std::vector<BigInt> v when computing the factor base.
FactorBase(CFB) returns the factor base obtained so far (as vector<BigInt>).
These functions are in NumTheory-prime. The functions
NextPrime, PrevPrime, RandomNBitPrime and RandomSmallPrime each produce a
result of type SmallPrime (essentially a long which is known
to be prime).
eratosthenes(n) build vector<bool> sieve of Eratosthenes up to n; entry k corresponds to integer 2*k+1; max valid index is n/2
EratosthenesRange(lo, hi) build vector<bool> sieve of Eratosthenes from lo up to hi; if lo is odd, it is replaced by lo+1; similarly for hi. In returned vector entry k corresponds to integer lo+2*k; max valid index is (hi-lo)/2
IsPrime(n) tests the positive number n for primality (may be very slow for larger numbers). Gives error if n <= 0.
IsProbPrime(n) tests the positive number n for primality (fairly fast for large numbers, but in very rare cases may falsely declare a number to be prime). Gives error if n <= 0.
IsProbPrime(n,iters) tests the positive number n for primality; performs iters iterations of the Miller-Rabin test (default value is 25). Gives error if n <= 0.
NextPrime(n) and PrevPrime(n) compute next or previous positive prime (fitting into a machine long). NextPrime returns 0 if no next "small" prime exists; PrevPrime throws OutOfRange if arg is less than 3. Both throw BadArg if n < 0.
RandomSmallPrime(N) -- generate a (uniform) random prime from 5 up to N; error if N < 5 or N >= 2^31. Result is of type SmallPrime (essentially a long).
RandomNBitPrime(N) -- generate a (uniform) random prime in range 2^(N-1) to 2^N; error if N < 10 or N >= 31. Result is of type SmallPrime (essentially a long).
NextProbPrime(N) and PrevProbPrime(N) compute next or previous positive probable prime (uses IsProbPrime). PrevProbPrime throws OutOfRange error if 0 <= N < 3. Both throw BadArg error if N < 0.
NextProbPrime(N,iters) and PrevProbPrime(N,iters) compute next or previous positive probable prime (uses IsProbPrime with second arg iters). PrevProbPrime throws OutOfRange error if 0 <= N < 3. Both throw BadArg error if N < 0.
There are also iterators for generating primes (or almost primes) in increasing order:
PrimeSeq() the sequence of primes starting with 2.
PrimeSeq1ModN(n) the sequence of primes 1 modulo n
PrimeSeqForCRT() a sequence of primes starting with some "large" value, and going upwards.
FastFinitePrimeSeq() a sequence containing all primes up to some limit (much faster than PrimeSeq); limit is given by mem fn myLastPrime.
FastMostlyPrimeSeq() a sequence containing all primes and a few composites (much faster than PrimeSeq).
NoSmallFactorSeq() a sequence of positive integers which have no small factors.
If pseq is one of these iterator objects then
*pseq gives the current prime in the sequence (as a value of type SmallPrime, or of type long for FastMostlyPrimeSeq and NoSmallFactorSeq)
++pseq advances 1 step along the sequence
IsEnded(pseq) returns true if the end of the sequence has been reached
CurrPrime(pseq) same as *pseq (only for PrimeSeq and PrimeSeqForCRT)
NextPrime(pseq) advances 1 step along the sequence, and returns the new "current prime" (only for PrimeSeq and PrimeSeqForCRT)
pseq.JumpTo(n) advance to n or beyond (will rewind if n is smaller than current value)
factor(n) finds the complete factorization of n (WARNING may be very slow for large numbers); NB implementation incomplete
factor_TrialDiv(n,limit) finds small prime factors of n (up to & including the specified limit); result is a factorization object. Gives error if limit is not positive or too large to fit into a long. See also MultiplicityOf2 in BigIntOps.
factor_PollardRho(n,niters) attempt to find a (single) factor of n (not nec. prime) using at most niters iterations; returns "empty" factorization if no factor was found; factor found may not be prime.
AllFactors(n) a vector<long> containing all positive factors of n in increasing; error if n <= 0
SumOfFactors(n,k) compute sum of k-th powers of positive factors of n
SmallestNonDivisor(n) finds smallest (positive prime) non-divisor of n; if n=0 throws ERR::NotNonZero.
IsSqFree(n) returns true if n is square-free, otherwise false; for BigInt result is a bool3
FactorMultiplicity(b,n) find largest k such that power(b,k) divides n (error if b < 2 or n is zero)
CoprimeFactor(N,b) effectively N/gcd(N,power(b,INFINITY)); for the "odd part" just compute CoprimeFactor(N,2)
PollardRhoSeq(N, start, incr) create a sequence starting from start and with increment incr
PRS.myAdvance(k) advance the sequence by k steps
GetFactor(PRS) returns a factor of N (may be 1 or N)
GetNumIters(PRS) returns number of steps/iters performed
EulerTotient(n) computes Euler's totient function of the positive number n (i.e. the number of integers up to n which are coprime to n, or the degree of the n-th cyclotomic polynomial). Gives error if n <= 0.
InvTotientBound(n) -- returns a BigInt representing an upper bound for the inverse EulerTotient of n -- using OEIS sequence A355667.
InvTotientBound_ulong(n) -- returns an unsigned long representing an upper bound for the inverse EulerTotient of n
InvTotientBoundUpto(n) -- returns a BigInt representing an upper bound for the inverse EulerTotient of all k <= n -- related to OEIS sequence A355667.
InvTotient(n) -- returns a vector<long> containing all possible preimages of n.
InvTotient(n, InvTotientMode::SqFreePreimages) -- returns a vector<long> containing all possible square-free preimages of n.
MoebiusFn(n) computes the Moebius function value of the positive number n (i.e. the sum of the primitive n-th roots of unity). Gives error if n <= 0.
PrimitiveRoot(p) computes the least positive primitive root for the positive prime p. Gives error if p is not a positive prime. WARNING May be very slow for large p (because it must factorize p-1).
KroneckerSymbol(res,mod) (test if res is a quadratic residue) computes the Kronecker symbol, generalization of Jacobi symbol, generalization of Legendre symbol
MultiplicativeOrderMod(res,mod) computes multiplicative order of res modulo mod. Throws error ERR::BadArg if mod < 2 or gcd(res,mod) is not 1.
PowerMod(base,exp,modulus) computes base to the power exp modulo modulus; result is least non-negative residue. If modulus is a machine integer then the result is of type long (or error if it does not fit), otherwise the result is of type BigInt. Gives error if modulus <= 1. Gives ERR::DivByZero if exp is negative and base cannot be inverted. If base and exp are both zero, it produces 1.
BinomialRepr(N,r) produces the repr of N as a sum of binomial coeffs with "denoms" r, r-1, r-2, ...
SimplestBigRatBetween(A,B) computes the simplest rational between A and B
SimplestBinaryRatBetween(A,B) computes the simplest binary rational between A and B; result is a rational of form N*2^k where the integer N is minimal.
Several of these functions give errors if they are handed unsuitable values:
unless otherwise indicated below the error is of type ERR::BadArg.
Recall that any real number has an expansion as a continued fraction (e.g. see Hardy & Wright for definition and many properties). This expansion is finite for any rational number. We adopt the following conventions which guarantee that the expansion is unique:
For example, with these conventions the expansion of -7/3 is (-3, 1, 2).
The main functions available are:
ContFracIter(q) constructs a new continued fraction iterator object
IsEnded(CFIter) true iff the iterator has moved past the last partial quotient
IsFinal(CFIter) true iff the iterator is at the last partial quotient
quot(CFIter) gives the current partial quotient as a BigInt (or throws ERR::IterEnded)
*CFIter gives the current partial quotient as a BigInt (or throws ERR::IterEnded)
++CFIter moves to next partial quotient (or throws ERR::IterEnded)
ContFracApproximant() for constructing a rational from its continued fraction quotients
CFA.myAppendQuot(q) appends the quotient q to the continued fraction
CFA.myRational() returns the rational associated to the continued fraction
CFApproximantsIter(q) constructs a new continued fraction approximant iterator
IsEnded(CFAIter) true iff the iterator has moved past the last "partial quotient"
*CFAIter gives the current continued fraction approximant as a BigRat (or throws ERR::IterEnded)
++CFAIter moves to next approximant (or throws ERR::IterEnded)
CFApprox(q,eps) gives the simplest cont. frac. approximant to q with relative error at most eps
CoCoALib offers the class CRTMill for reconstructing an integer from
several residue-modulus pairs via Chinese Remaindering. At the moment the
moduli from distinct pairs must be coprime.
The operations available are:
CRTMill() ctor; initially the residue is 0 and the modulus is 1
CRT.myAddInfo(res,mod) give a new residue-modulus pair to the CRTMill (error if mod is not coprime to all previous moduli)
CRT.myAddInfo(res,mod,CRTMill::CoprimeModulus) give a new residue-modulus pair to the CRTMill asserting that mod is coprime to all previous moduli -- CRTMill::CoprimeModulus is a constant
CombinedResidue(CRT) the combined residue with absolute value less than or equal to CombinedModulus(CRT)/2
CombinedModulus(CRT) the product of the moduli of all pairs given to the mill
CoCoALib offers two heuristic methods for reconstructing rationals from residue-modulus pairs; they have the same user interface but internally one algorithm is based on continued fractions while the other uses lattice reduction. The methods are heuristic, so may (rarely) produce an incorrect result.
NOTE the heuristic requires the (combined) modulus to be a certain amount larger than strictly necessary to reconstruct the correct answer (assuming perfect bounds are known). In practice, this means that the methods always fail if the combined modulus is too small.
The constructors available are:
RatReconstructByContFrac() ctor for continued fraction method mill log-epsilon equal to 20
RatReconstructByContFrac(LogEps) ctor for continued fraction method mill with given log-epsilon (must be at least 3)
RatReconstructByLattice(SafetyFactor) ctor for lattice method mill with given SafetyFactor (0 --> use default)
The operations available are:
reconstructor.myAddInfo(res,mod) give a new residue-modulus pair to the reconstructor
IsConvincing(reconstructor) gives true iff the mill can produce a convincing result
ReconstructedRat(reconstructor) gives the reconstructed rational (or an error if IsConvincing is not true).
BadMFactor(reconstructor) gives the "bad factor" of the combined modulus.
There is also a function for deterministic rational reconstruction which requires certain bounds to be given in input. It uses the continued fraction method.
RatReconstructWithBounds(e,P,Q,res,mod) where e is upper bound for number of "bad" moduli, P and Q are upper bounds for numerator and denominator of the rational to be reconstructed, and (res[i],mod[i]) is a residue-modulus pair with distinct moduli being coprime.
ExtendedEuclideanAlg is not immediately clear,
because the cofactor variables could conceivably overflow -- in fact
this cannot happen (at least on a binary computer): for a proof see
Shoup's book A Computational Introduction to Number Theory and Algebra,
in particular Theorem 4.3 and the comment immediately following it. There is
just one line where a harmless "overflow" could occur -- it is commented in
the code.
ExtGcd give an error if the inputs are both zero
because this is an exceptional case, and so should be handled specially. I
note that mpz_exgcd accepts this case, and returns two zero cofactors; so if we
decide to accept this case, we should do the same -- this all fits in well with
the (reasonable/good) principle that "zero inputs have zero cofactors".
- Several functions are more complicated than you might expect because I wanted them
to be correct for all possible machine integer inputs (e.g. including the
most negative long value).
unsigned long values: the function should
normally be used only via the "dispatch" functions whose args are of type
MachineInt or BigInt.
NumTheory-prime.
eratosthenes is fairly straightforward given that I chose
to represent only odd numbers in the table: the k-th entry corresponds
to the integer 2*k+1. Overflow cannot occur: recall that the table
size is at most half the biggest long. I'm hoping that C++11 will
avoid the cost of copying the result upon returning. Anyway, I think the
code is a decent compromise between readability, speed and space efficiency.
The impl of EratosthenesRange is similar but the table covers just
the given range (only odd numbers are represented; index 0 corresponds to
smallest odd integer greater than or equal to the start of the range).
NoSmallFactorSeq is somewhat arbitrary. Not sure the code is
32-bit safe.
ContFracIter is represented by both
myFrac and myQuot being zero. This means that a newly created
iterator for zero is already ended.
CFApproximantsIter delegates most of the work to ContFracIter.
long values when perhaps unsigned long
would possibly be better choice (since it offers a greater range, and
in the case of gcd it would permit the fn to return a result always,
rather than report "overflow"). The choice of return type was dictated
by the coding conventions, which were in turn dictated by the risks of nasty
surprises to unwary users unfamiliar with the foibles of unsigned values in C++.
NextPrime has dodgy semantics: what happens when the end of the
iterator is reached? In fact, all the non-finite "prime seq" iterators
do not handle end-of-iterator properly!
BigInt values?
(e.g. gcd, lcm, InvMod, PowerMod, and so on). (2016-06-27 this will probably become irrelevant when using "move" semantics in C++11).
PowerMod should be improved (e.g. to use
PowerModSmallModulus whenever possible). Is behaviour for 0^0 correct?
KroneckerSymbol I have chosen to make available just KroneckerSymbol
rather than the more widely-known LegendreSymbol because GMP makes
Kronecker available, and it is always defined; whereas
LegendreSymbol would have to check that its 2nd arg is a prime
(which would dominate the cost of the call)
LucasTest should produce a certificate, and be made publicly accessible.
ContFracIter could be rather more efficient for rationals having
very large numerator and denominator. One way would be to compute with
num and den divided by the same large factor (probably a power of 2),
and taking care to monitor how accurate these "scaled" num and den are.
I'll wait until there is a real need before implementing (as I expect
it will turn out a bit messy).
CFApproximantsIter::operator++() should be made more efficient.
2022
SmoothFactor has been renamed factor_TrialDiv
factor_PollardRho
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