Number Theory
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## Number Theory

AreRelativelyPrime

      AreRelativelyPrime (a,b)

  Are the real integers `a` and `b` relatively prime? Returns `true` or `false`.

  See Wikipedia or Planetmath or Mathworld for more information.

BernoulliNumber

      BernoulliNumber (n)

  Return the `n`th Bernoulli number.

  See Wikipedia or Mathworld for more information.

ChineseRemainder

      ChineseRemainder (a,m)

  Aliases: `CRT`

  Find the `x` that solves the system given by the vector `a` and modulo the elements of `m`, using the Chinese Remainder Theorem.

  See Wikipedia or Planetmath or Mathworld for more information.

CombineFactorizations

      CombineFactorizations (a,b)

  Given two factorizations, give the factorization of the product.

  See Factorize.

ConvertFromBase

      ConvertFromBase (v,b)

  Convert a vector of values indicating powers of b to a number.

ConvertToBase

      ConvertToBase (n,b)

  Convert a number to a vector of powers for elements in base `b`.

DiscreteLog

      DiscreteLog (n,b,q)

  Find discrete log of `n` base `b` in Fq, the finite field of order `q`, where `q` is a prime, using the Silver-Pohlig-Hellman algorithm.

  See Wikipedia, Planetmath, or Mathworld for more information.

Divides

      Divides (m,n)

  Checks divisibility (if `m` divides `n`).

EulerPhi

      EulerPhi (n)

  Compute the Euler phi function for `n`, that is the number of integers between 1 and `n` relatively prime to `n`.

  See Wikipedia, Planetmath, or Mathworld for more information.

ExactDivision

      ExactDivision (n,d)

  Return `n/d` but only if `d` divides `n`. If `d` does not divide `n` then this function returns garbage. This is a lot faster for very large numbers than the operation `n/d`, but it is only useful if you know that the division is exact.

Factorize

      Factorize (n)

  Return factorization of a number as a matrix. The first row is the primes in the factorization (including 1) and the second row are the powers. So for example:

      genius> Factorize(11*11*13)
      =
      [1      11      13
       1      2       1]

  See Wikipedia for more information.

Factors

      Factors (n)

  Return all factors of `n` in a vector. This includes all the non-prime factors as well. It includes 1 and the number itself. So to print all the perfect numbers (those that are sums of their factors) up to the number 1000 you could do (this is clearly very inefficient)

      for n=1 to 1000 do (
          if MatrixSum (Factors(n)) == 2*n then
              print(n)
      )

FermatFactorization

      FermatFactorization (n,tries)

  Attempt Fermat factorization of `n` into `(t-s)*(t+s)`, returns `t` and `s` as a vector if possible, `null` otherwise. `tries` specifies the number of tries before giving up.

  This is a fairly good factorization if your number is the product of two factors that are very close to each other.

  See Wikipedia for more information.

FindPrimitiveElementMod

      FindPrimitiveElementMod (q)

  Find the first primitive element in Fq, the finite group of order `q`. Of course `q` must be a prime.

FindRandomPrimitiveElementMod

      FindRandomPrimitiveElementMod (q)

  Find a random primitive element in Fq, the finite group of order `q` (q must be a prime).

IndexCalculus

      IndexCalculus (n,b,q,S)

  Compute discrete log base `b` of n in Fq, the finite group of order `q` (`q` a prime), using the factor base `S`. `S` should be a column of primes possibly with second column precalculated by `IndexCalculusPrecalculation`.

IndexCalculusPrecalculation

      IndexCalculusPrecalculation (b,q,S)

  Run the precalculation step of `IndexCalculus` for logarithms base `b` in Fq, the finite group of order `q` (`q` a prime), for the factor base `S` (where `S` is a column vector of primes). The logs will be precalculated and returned in the second column.

IsEven

      IsEven (n)

  Tests if an integer is even.

IsMersennePrimeExponent

      IsMersennePrimeExponent (p)

  Tests if a positive integer `p` is a Mersenne prime exponent. That is if 2p-1 is a prime. It does this by looking it up in a table of known values, which is relatively short. See also MersennePrimeExponents and LucasLehmer.

  See Wikipedia, Planetmath, Mathworld or GIMPS for more information.

IsNthPower

      IsNthPower (m,n)

  Tests if a rational number `m` is a perfect `n`th power. See also IsPerfectPower and IsPerfectSquare.

IsOdd

      IsOdd (n)

  Tests if an integer is odd.

IsPerfectPower

      IsPerfectPower (n)

  Check an integer for being any perfect power, ab.

IsPerfectSquare

      IsPerfectSquare (n)

  Check an integer for being a perfect square of an integer. The number must be an integer. Negative integers are never perfect squares of integers.

IsPrime

      IsPrime (n)

  Tests primality of integers, for numbers less than 2.5e10 the answer is deterministic (if Riemann hypothesis is true). For numbers larger, the probability of a false positive depends on `IsPrimeMillerRabinReps`. That is the probability of false positive is 1/4 to the power `IsPrimeMillerRabinReps`. The default value of 22 yields a probability of about 5.7e-14.

  If `false` is returned, you can be sure that the number is a composite. If you want to be absolutely sure that you have a prime you can use `MillerRabinTestSure` but it may take a lot longer.

  See Planetmath or Mathworld for more information.

IsPrimitiveMod

      IsPrimitiveMod (g,q)

  Check if `g` is primitive in Fq, the finite group of order `q`, where `q` is a prime. If `q` is not prime results are bogus.

IsPrimitiveModWithPrimeFactors

      IsPrimitiveModWithPrimeFactors (g,q,f)

  Check if `g` is primitive in Fq, the finite group of order `q`, where `q` is a prime and `f` is a vector of prime factors of `q`-1. If `q` is not prime results are bogus.

IsPseudoprime

      IsPseudoprime (n,b)

  If `n` is a pseudoprime base `b` but not a prime, that is if `b^(n-1) == 1 mod n`. This calls the `PseudoprimeTest`

IsStrongPseudoprime

      IsStrongPseudoprime (n,b)

  Test if `n` is a strong pseudoprime to base `b` but not a prime.

Jacobi

      Jacobi (a,b)

  Aliases: `JacobiSymbol`

  Calculate the Jacobi symbol (a/b) (b should be odd).

JacobiKronecker

      JacobiKronecker (a,b)

  Aliases: `JacobiKroneckerSymbol`

  Calculate the Jacobi symbol (a/b) with the Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even.

LeastAbsoluteResidue

      LeastAbsoluteResidue (a,n)

  Return the residue of `a` mod `n` with the least absolute value (in the interval -n/2 to n/2).

Legendre

      Legendre (a,p)

  Aliases: `LegendreSymbol`

  Calculate the Legendre symbol (a/p).

  See Planetmath or Mathworld for more information.

LucasLehmer

      LucasLehmer (p)

  Test if 2p-1 is a Mersenne prime using the Lucas-Lehmer test. See also MersennePrimeExponents and IsMersennePrimeExponent.

  See Wikipedia, Planetmath, or Mathworld for more information.

LucasNumber

      LucasNumber (n)

  Returns the `n`th Lucas number.

  See Wikipedia, Planetmath, or Mathworld for more information.

MaximalPrimePowerFactors

      MaximalPrimePowerFactors (n)

  Return all maximal prime power factors of a number.

MersennePrimeExponents

      MersennePrimeExponents

  A vector of known Mersenne prime exponents, that is a list of positive integers `p` such that 2p-1 is a prime. See also IsMersennePrimeExponent and LucasLehmer.

  See Wikipedia, Planetmath, Mathworld or GIMPS for more information.

MillerRabinTest

      MillerRabinTest (n,reps)

  Use the Miller-Rabin primality test on `n`, `reps` number of times. The probability of false positive is `(1/4)^reps`. It is probably usually better to use `IsPrime` since that is faster and better on smaller integers.

  See Wikipedia or Planetmath or Mathworld for more information.

MillerRabinTestSure

      MillerRabinTestSure (n)

  Use the Miller-Rabin primality test on `n` with enough bases that assuming the Generalized Riemann Hypothesis the result is deterministic.

  See Wikipedia, Planetmath, or Mathworld for more information.

ModInvert

      ModInvert (n,m)

  Returns inverse of n mod m.

  See Mathworld for more information.

MoebiusMu

      MoebiusMu (n)

  Return the Moebius mu function evaluated in `n`. That is, it returns 0 if `n` is not a product of distinct primes and `(-1)^k` if it is a product of `k` distinct primes.

  See Planetmath or Mathworld for more information.

NextPrime

      NextPrime (n)

  Returns the least prime greater than `n`. Negatives of primes are considered prime and so to get the previous prime you can use `-NextPrime(-n)`.

  This function uses the GMPs `mpz_nextprime`, which in turn uses the probabilistic Miller-Rabin test (See also `MillerRabinTest`). The probability of false positive is not tunable, but is low enough for all practical purposes.

  See Planetmath or Mathworld for more information.

PadicValuation

      PadicValuation (n,p)

  Returns the p-adic valuation (number of trailing zeros in base `p`).

  See Wikipedia or Planetmath for more information.

PowerMod

      PowerMod (a,b,m)

  Compute `a^b mod m`. The `b`'s power of `a` modulo `m`. It is not necessary to use this function as it is automatically used in modulo mode. Hence `a^b mod m` is just as fast.

Prime

      Prime (n)

  Aliases: `prime`

  Return the `n`th prime (up to a limit).

  See Planetmath or Mathworld for more information.

PrimeFactors

      PrimeFactors (n)

  Return all prime factors of a number as a vector.

  See Wikipedia or Mathworld for more information.

PseudoprimeTest

      PseudoprimeTest (n,b)

  Pseudoprime test, returns `true` if and only if `b^(n-1) == 1 mod n`

  See Planetmath or Mathworld for more information.

RemoveFactor

      RemoveFactor (n,m)

  Removes all instances of the factor `m` from the number `n`. That is divides by the largest power of `m`, that divides `n`.

  See Planetmath or Mathworld for more information.

SilverPohligHellmanWithFactorization

      SilverPohligHellmanWithFactorization (n,b,q,f)

  Find discrete log of `n` base `b` in Fq, the finite group of order `q`, where `q` is a prime using the Silver-Pohlig-Hellman algorithm, given `f` being the factorization of `q`-1.

SqrtModPrime

      SqrtModPrime (n,p)

  Find square root of `n` modulo `p` (where `p` is a prime). Null is returned if not a quadratic residue.

  See Planetmath or Mathworld for more information.

StrongPseudoprimeTest

      StrongPseudoprimeTest (n,b)

  Run the strong pseudoprime test base `b` on `n`.

  See Wikipedia, Planetmath, or Mathworld for more information.

gcd

      gcd (a,args...)

  Aliases: `GCD`

  Greatest common divisor of integers. You can enter as many integers as you want in the argument list, or you can give a vector or a matrix of integers. If you give more than one matrix of the same size then GCD is done element by element.

  See Wikipedia, Planetmath, or Mathworld for more information.

lcm

      lcm (a,args...)

  Aliases: `LCM`

  Least common multiplier of integers. You can enter as many integers as you want in the argument list, or you can give a vector or a matrix of integers. If you give more than one matrix of the same size then LCM is done element by element.

  See Wikipedia, Planetmath, or Mathworld for more information.

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