# Integer and Number Theoretic Functions

Mod[k,n] | k modulo n (remainder from dividing k by n) |

Quotient[m,n] | the quotient of m and n (truncation of ) |

QuotientRemainder[m,n] | a list of the quotient and the remainder |

Divisible[m,n] | test whether m is divisible by n |

CoprimeQ[n_{1},n_{2},...] | test whether the are pairwise relatively prime |

GCD[n_{1},n_{2},...] | the greatest common divisor of , , ... |

LCM[n_{1},n_{2},...] | the least common multiple of , , ... |

KroneckerDelta[n_{1},n_{2},...] | the Kronecker delta equal to 1 if all the are equal, and 0 otherwise |

IntegerDigits[n,b] | the digits of n in base b |

IntegerExponent[n,b] | the highest power of b that divides n |

Some integer functions.

The remainder on dividing 17 by 3.

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The integer part of 17/3.

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Mod also works with real numbers.

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The result from

Mod always has the same sign as the second argument.

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For any integers

a and

b, it is always true that

b*Quotient[a, b]+Mod[a, b] is equal to

.

Mod[k,n] | result in the range 0 to |

Mod[k,n,1] | result in the range 1 to n |

Mod[k,n,-n/2] | result in the range to |

Mod[k,n,d] | result in the range d to |

Integer remainders with offsets.

Particularly when you are using

Mod to get indices for parts of objects, you will often find it convenient to specify an offset.

This effectively extracts the 18

part of the list, with the list treated cyclically.

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The greatest common divisor function

GCD gives the largest integer that divides all the

exactly. When you enter a ratio of two integers,

*Mathematica* effectively uses

GCD to cancel out common factors and give a rational number in lowest terms.

The least common multiple function

LCM gives the smallest integer that contains all the factors of each of the

.

The largest integer that divides both 24 and 15 is 3.

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The Kronecker delta function

KroneckerDelta is equal to 1 if all the

are equal, and is 0 otherwise.

can be thought of as a totally symmetric tensor.

This gives a totally symmetric tensor of rank 3.

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Integer factoring and related functions.

This gives the factors of 24 as

,

. The first element in each list is the factor; the second is its exponent.

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Here are the factors of a larger integer.

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You should realize that according to current mathematical thinking, integer factoring is a fundamentally difficult computational problem. As a result, you can easily type in an integer that

*Mathematica* will not be able to factor in anything short of an astronomical length of time. But as long as the integers you give are less than about 50 digits long,

FactorInteger should have no trouble. And in special cases it will be able to deal with much longer integers.

Here is a rather special long integer.

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*Mathematica* can easily factor this special integer.

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Although

*Mathematica* may not be able to factor a large integer, it can often still test whether or not the integer is a prime. In addition,

*Mathematica* has a fast way of finding the

prime number.

It is often much faster to test whether a number is prime than to factor it.

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Here is a plot of the first 100 primes.

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This is the millionth prime.

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Particularly in number theory, it is often more important to know the distribution of primes than their actual values. The function

PrimePi[x] gives the number of primes

that are less than or equal to

.

This gives the number of primes less than a billion.

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PrimeNu gives the number of distinct primes.

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PrimeOmega gives the number of prime factors counting multiplicities

in

n.

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Liouville's function gives

where

is the number of prime factors counting multiplicity.

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The Mangoldt function returns the log of prime power base or zero when composite.

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By default,

FactorInteger allows only real integers. But with the option setting

GaussianIntegers->True, it also handles Gaussian integers, which are complex numbers with integer real and imaginary parts. Just as it is possible to factor uniquely in terms of real primes, it is also possible to factor uniquely in terms of Gaussian primes. There is nevertheless some potential ambiguity in the choice of Gaussian primes. In

*Mathematica*, they are always chosen to have positive real parts, and non-negative imaginary parts, except for a possible initial factor of

or

.

Over the Gaussian integers, 2 can be factored as

.

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Here are the factors of a Gaussian integer.

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PowerMod[a,b,n] | the power modulo n |

DirichletCharacter[k,j,n] | the Dirichlet character |

EulerPhi[n] | the Euler totient function |

MoebiusMu[n] | the Möbius function |

DivisorSigma[k,n] | the divisor function |

DivisorSum[n,form] | the sum of for all i that divide n |

DivisorSum[n,form,cond] | the sum for only those divisors for which gives True |

JacobiSymbol[n,m] | the Jacobi symbol |

ExtendedGCD[n_{1},n_{2},...] | the extended GCD of , , ... |

MultiplicativeOrder[k,n] | the multiplicative order of k modulo n |

MultiplicativeOrder[k,n,{r_{1},r_{2},...}] | the generalized multiplicative order with residues |

CarmichaelLambda[n] | the Carmichael function |

PrimitiveRoot[n] | a primitive root of n |

Some functions from number theory.

The modular power function

PowerMod gives exactly the same results as

Mod for

b>0.

PowerMod is much more efficient, however, because it avoids generating the full form of

.

You can use

PowerMod not only to find positive modular powers, but also to find modular inverses. For negative

b,

PowerMod gives, if possible, an integer

such that

. (Whenever such an integer exists, it is guaranteed to be unique modulo

n.) If no such integer

exists,

*Mathematica* leaves

PowerMod unevaluated.

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This gives the modular inverse of 3 modulo 7.

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Multiplying the inverse by 3 modulo 7 gives 1, as expected.

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This finds the smallest non-negative integer

so that

is equal to 3 mod 11.

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This verifies the result.

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This returns all integers less than 11 which satisfy the relation.

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If

d does not have a square root modulo

n,

PowerMod will remain unevaluated and

PowerModList will return an empty list.

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This checks that 3 is not a square modulo 5.

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Even for a large modulus, the square root can be computed fairly quickly.

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There are

distinct Dirichlet characters for a given modulus

k, as labeled by the index

j. Different conventions can give different orderings for the possible characters.

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The Euler totient function

gives the number of integers less than

that are relatively prime to

. An important relation (Fermat's little theorem) is that

for all

relatively prime to

.

The Möbius function

is defined to be

if

is a product of

distinct primes, and

if

contains a squared factor (other than 1). An important relation is the Möbius inversion formula, which states that if

for all

, then

, where the sums are over all positive integers

that divide

.

The divisor function

is the sum of the

powers of the divisors of

. The function

gives the total number of divisors of

, and is variously denoted

,

and

. The function

, equal to the sum of the divisors of

, is often denoted

.

For prime

,

.

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The result is 1, as guaranteed by Fermat's little theorem.

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This gives a list of all the divisors of 24.

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gives the total number of distinct divisors of 24.

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The function

DivisorSum represents the sum of

for all

i that divide

n.

DivisorSum includes only those divisors for which

gives

True.

This gives a list of sums for the divisors of five positive integers.

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This imposes the condition that the value of each divisor

i must be less than 6.

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The Jacobi symbol

JacobiSymbol reduces to the Legendre symbol

when

is an odd prime. The Legendre symbol is equal to zero if

is divisible by

; otherwise it is equal to

if

is a quadratic residue modulo the prime

, and to

if it is not. An integer

relatively prime to

is said to be a quadratic residue modulo

if there exists an integer

such that

. The full Jacobi symbol is a product of the Legendre symbols

for each of the prime factors

such that

.

The extended GCD

ExtendedGCD gives a list

where

is the greatest common divisor of the

, and the

are integers such that

. The extended GCD is important in finding integer solutions to linear Diophantine equations.

The first number in the list is the GCD of 105 and 196.

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The second pair of numbers satisfies

.

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The multiplicative order function

MultiplicativeOrder gives the smallest integer

such that

. Then

is known as the order of

modulo

. The notation

is occasionally used.

The generalized multiplicative order function

MultiplicativeOrder gives the smallest integer

such that

for some

.

MultiplicativeOrder is sometimes known as the suborder function of

modulo

, denoted

.

MultiplicativeOrder is sometimes known as the discrete log of

with respect to the base

modulo

.

The Carmichael function or least universal exponent

gives the smallest integer

such that

for all integers

relatively prime to

.

ContinuedFraction[x,n] | generate the first n terms in the continued fraction representation of x |

FromContinuedFraction[list] | reconstruct a number from its continued fraction representation |

Rationalize[x,dx] | find a rational approximation to x with tolerance dx |

Continued fractions.

This generates the first 10 terms in the continued fraction representation for

.

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This reconstructs the number represented by the list of continued fraction terms.

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The result is close to

.

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This gives directly a rational approximation to

.

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Continued fractions appear in many number theoretic settings. Rational numbers have terminating continued fraction representations. Quadratic irrational numbers have continued fraction representations that become repetitive.

ContinuedFraction[x] | the complete continued fraction representation for a rational or quadratic irrational number |

QuadraticIrrationalQ[x] | test whether x is a quadratic irrational |

RealDigits[x] | the complete digit sequence for a rational number |

RealDigits[x,b] | the complete digit sequence in base b |

Complete representations for numbers.

The continued fraction representation of

starts with the term 8, then involves a sequence of terms that repeat forever.

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This reconstructs

from its continued fraction representation.

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This number is a quadratic irrational.

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This shows the recurring sequence of decimal digits in 3/7.

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Continued fraction convergents are often used to approximate irrational numbers by rational ones. Those approximations alternate from above and below, and converge exponentially in the number of terms. Furthermore, a convergent

of a simple continued fraction is better than any other rational approximation with denominator less than or equal to

.

Convergents[x] | give a list of rational approximations of x |

Convergents[x,n] | give only the first n approximations |

Continued fraction convergents.

This gives a list of rational approximations of 101/9801, derived from its continued fraction expansion.

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This lists only the first 10 convergents.

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This lists successive rational approximations to

, until the numerical precision is exhausted.

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With an exact irrational number, you have to explicitly ask for a certain number of terms.

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LatticeReduce[{v_{1}v_{2},...}] | a reduced lattice basis for the set of integer vectors |

HermiteDecomposition[{v_{1},v_{2},...}] | the echelon form for the set of integer vectors |

Functions for integer lattices.

The lattice reduction function

LatticeReduce is used in several kinds of modern algorithms. The basic idea is to think of the vectors

of integers as defining a mathematical

*lattice*. Any vector representing a point in the lattice can be written as a linear combination of the form

, where the

are integers. For a particular lattice, there are many possible choices of the "basis vectors"

. What

LatticeReduce does is to find a reduced set of basis vectors

for the lattice, with certain special properties.

Three unit vectors along the three coordinate axes already form a reduced basis.

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This gives the reduced basis for a lattice in four-dimensional space specified by three vectors.

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Notice that in the last example,

LatticeReduce replaces vectors that are nearly parallel by vectors that are more perpendicular. In the process, it finds some quite short basis vectors.

For a matrix

,

HermiteDecomposition gives matrices

and

such that

is unimodular,

, and

is in reduced row echelon form. In contrast to

RowReduce, pivots may be larger than 1 because there are no fractions in the ring of integers. Entries above a pivot are minimized by subtracting appropriate multiples of the pivot row.

In this case, the original matrix is recovered because it was in row echelon form.

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This satisfies the required identities.

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Here the second matrix has some pivots larger than 1, and nonzero entries over pivots.

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DigitCount[n,b,d] | the number of d digits in the base b representation of n |

Digit count function.

Here are the digits in the base 2 representation of the number 77.

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This directly computes the number of ones in the base 2 representation.

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The plot of the digit count function is self-similar.

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BitAnd[n_{1},n_{2},...] | bitwise AND of the integers |

BitOr[n_{1},n_{2},...] | bitwise OR of the integers |

BitXor[n_{1},n_{2},...] | bitwise XOR of the integers |

BitNot[n] | bitwise NOT of the integer n |

BitLength[n] | number of binary bits in the integer n |

BitSet[n,k] | set bit k to 1 in the integer n |

BitGet[n,k] | get bit k from the integer n |

BitClear[n,k] | set bit k to 0 in the integer n |

BitShiftLeft[n,k] | shift the integer n to the left by k bits, padding with zeros |

BitShiftRight[n,k] | shift to the right, dropping the last k bits |

Bitwise operations.

Bitwise operations act on integers represented as binary bits.

BitAnd yields the integer whose binary bit representation has ones at positions where the binary bit representations of all of the

have ones.

BitOr yields the integer with ones at positions where any of the

have ones.

BitXor yields the integer with ones at positions where

or

but not both have ones.

BitXor has ones where an odd number of the

have ones.

This finds the bitwise AND of the numbers 23 and 29 entered in base 2.

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| |

Bitwise operations are used in various combinatorial algorithms. They are also commonly used in manipulating bitfields in low-level computer languages. In such languages, however, integers normally have a limited number of digits, typically a multiple of 8. Bitwise operations in

*Mathematica* in effect allow integers to have an unlimited number of digits. When an integer is negative, it is taken to be represented in two's complement form, with an infinite sequence of ones on the left. This allows

BitNot[n] to be equivalent simply to

.

Testing for a squared factor.

SquareFreeQ[n] checks to see if

n has a square prime factor. This is done by computing

MoebiusMu[n] and seeing if the result is zero; if it is, then

n is not squarefree, otherwise it is. Computing

MoebiusMu[n] involves finding the smallest prime factor

q of

n. If

n has a small prime factor (less than or equal to

), this is very fast. Otherwise,

FactorInteger is used to find

q.

This product of primes contains no squared factors.

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The square number 4 divides 60.

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NextPrime[n] | give the smallest prime larger than n |

RandomPrime[{min,max}] | return a random prime number between min and max |

RandomPrime[max] | return a random prime number less than or equal to max |

RandomPrime[{min,max},n] | return n random prime numbers between min and max |

RandomPrime[max,n] | return n random prime numbers less than or equal to max |

Finding prime numbers.

NextPrime[n] finds the smallest prime

p such that

. For

n less than 20 digits, the algorithm does a direct search using

PrimeQ on the odd numbers greater than

n. For

n with more than 20 digits, the algorithm builds a small sieve and first checks to see whether the candidate prime is divisible by a small prime before using

PrimeQ. This seems to be slightly faster than a direct search.

This gives the next prime after 10.

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Even for large numbers, the next prime can be computed rather quickly.

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This gives the largest prime less than 34.

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For

RandomPrime and

RandomPrime[max], a random prime

p is obtained by randomly selecting from a prime lookup table if

max is small and by a random search of integers in the range if

max is large. If no prime exists in the specified range, the input is returned unevaluated with an error message.

Here is a random prime between 10 and 100.

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PrimePowerQ[n] | determine whether n is a positive integer power of a rational prime |

Testing for involving prime powers.

The algorithm for

PrimePowerQ involves first computing the least prime factor

p of

n and then attempting division by

n until either 1 is obtained, in which case

n is a prime power, or until division is no longer possible, in which case

n is not a prime power.

Here is a number that is a power of a single prime.

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ChineseRemainder[list_{1},list_{2}] | give the smallest non-negative integer r with Mod[r, list_{2}]==list_{1} |

Solving simultaneous congruences.

The Chinese remainder theorem states that a certain class of simultaneous congruences always has a solution.

ChineseRemainder finds the smallest non-negative integer

r such that

Mod is

. The solution is unique modulo the least common multiple of the elements of

.

This means that

,

, and

.

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This confirms the result.

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Longer lists are still quite fast.

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PrimitiveRoot[n] | give a primitive root of n, where n is a prime power or twice a prime power |

Computing primitive roots.

PrimitiveRoot[n] returns a generator for the group of numbers relatively prime to

n under multiplication

. This has a generator if and only if

n is 2, 4, a power of an odd prime, or twice a power of an odd prime. If

n is a prime or prime power, the least positive primitive root will be returned.

Here is a primitive root of 5.

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This confirms that it does generate the group.

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Here is a primitive root of a prime power.

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Here is a primitive root of twice a prime power.

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If the argument is composite and not a prime power or twice a prime power, the function does not evaluate.

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SquaresR[d,n] | give the number of representations of an integer n as a sum of d squares |

PowersRepresentations[n,k,p] | give the distinct representations of the integer n as a sum of k non-negative p integer powers |

Representing an integer as a sum of squares or other powers.

Here are the representations of 101 as a sum of 3 squares.

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