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 (n)
Return the nth Bernoulli number.
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 (a,b)
Given two factorizations, give the factorization of the product.
See Factorize.
ConvertFromBase (v,b)
Convert a vector of values indicating powers of b to a number.
ConvertToBase (n,b)
Convert a number to a vector of powers for elements in base b.
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 (m,n)
Checks divisibility (if m divides n).
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 (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 of course only
	    useful if you know that the division is exact.
	  
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 (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 for example to print all the perfect numbers
	    (those that are sums of their factors) up to the number 1000 you
	    could do (this is of course very inefficient)
	    
for n=1 to 1000 do (
    if MatrixSum (Factors(n)) == 2*n then
        print(n)
)
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 (q)
Find the first primitive element in Fq, the finite
group of order q.  Of course q must be a prime.
FindRandomPrimitiveElementMod (q)
Find a random primitive element in Fq, the finite
group of order q (q must be a prime).
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 (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 (n)
Überprüft, ob eine Ganzzahl gerade ist.
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 (m,n)
	    Tests if a rational number m is a perfect
	    nth power.  See also
	    IsPerfectPower
	    and
	    IsPerfectSquare.
	  
IsOdd (n)
Überprüft, ob eine Ganzzahl ungerade ist.
IsPerfectPower (n)
Check an integer for being any perfect power, ab.
IsPerfectSquare (n)
Check an integer for being a perfect square of an integer. The number must be an integer. Negative integers are of course never perfect squares of integers.
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 (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 (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 (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 (n,b)
Test if n is a strong pseudoprime to base b but not a prime.
Jacobi (a,b)
Aliases: JacobiSymbol
Calculate the Jacobi symbol (a/b) (b should be odd).
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 (a,n)
Return the residue of a mod n with the least absolute value (in the interval -n/2 to n/2).
Legendre (a,p)
Aliases: LegendreSymbol
Calculate the Legendre symbol (a/p).
See Planetmath or Mathworld for more information.
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 (n)
Returns the nth Lucas number.
See Wikipedia, Planetmath, or Mathworld for more information.
MaximalPrimePowerFactors (n)
Return all maximal prime power factors of a number.
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 (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 (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 (n,m)
Returns inverse of n mod m.
See Mathworld for more information.
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 (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 (n,p)
Returns the p-adic valuation (number of trailing zeros in base p).
See Wikipedia or Planetmath for more information.
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 (n)
Aliases: prime
Return the nth prime (up to a limit).
See Planetmath or Mathworld for more information.
PrimeFactors (n)
Return all prime factors of a number as a vector.
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 (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 (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 (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 (n,b)
Run the strong pseudoprime test base b on n.
See Wikipedia, Planetmath, or Mathworld for more information.
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 (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.