# Miscellaneous number theory functions

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From: Link on 8 Apr 2010 15:27

---

# Miscellaneous number theory functions

function IsNthPower(m,n) =
(
if IsMatrix (m) or IsMatrix (n) then
return ApplyOverMatrix2 (m, n, IsNthPower)
else if not IsRational (m) then
# Only integers/rationals can be perfect powers
false
else if n == 2 then
IsPerfectSquare(m)
else if not IsPerfectPower(m) then
# This may not work
false
else if IsInteger(m) then
IsInteger(m^(1/n))

# Since rational numbers are in reduced form, by unique
factorization it
# suffices to check if the numerator and denominator are Nth powers
else
(IsNthPower(Numerator(m),n)) and IsNthPower(Denominator(m),n)
)
SetHelp("IsNthPower","number_theory","Tests if a rational number is a
perfect power");

# Returns the highest power of p that divides n, where n is rational.
# FIXME: This is only a valuation if p is prime, so this is an ab use
# of language.
function PadicValuation(n,p) =
(
if IsRational(n) and not IsInteger(n) then
return PadicValuation(Numerator(n))-
PadicValuation(Denominator(n));
if not IsInteger(n) then
(
error("PadicValuation: argument must be rational or integer");
bailout
);
# if (n==0) then return Infinity; # FIXME: Infinity not yet
implemented
if (n==0) then return null;

valuation=0;
while Divides(p,n) do
(
valuation=valuation+1;
n=ExactDivision(n,p);
);
valuation
)
SetHelp("PadicValuation","number_theory","Returns the padic valuation
(number of trailing zeros in base p).");

function RemoveFactor(n,m) =
(
if not IsInteger(n) or not IsInteger(m) then
(
error("RemoveFactor: arguments must be integers");
bailout
);
if (n==0) then return 0;

while Divides(m,n) do
(
n = ExactDivision(n,m);
);
n
)
SetHelp("RemoveFactor","number_theory","Removes all instances of the
factor m from the number n");

function PowerMod(a,b,m) = a^b mod m
SetHelp("PowerMod","number_theory","Compute a^b mod m");

function AreRelativelyPrime(a,b) = (
if IsMatrix (a) or IsMatrix (b) then
return ApplyOverMatrix2 (a, b, AreRelativelyPrime);
gcd(a,b) == 1
)
SetHelp("AreRelativelyPrime","number_theory","Are a and b relatively
prime?");

function LeastAbsoluteResidue(a,n) = (
if not IsInteger(a) or not IsPositiveInteger(n) then
(
error("LeastAbsoluteResidue: arguments must be integers");
bailout
);
b = a mod n;
if (-b+n) < b then
b-n
else
b
)
SetHelp("LeastAbsoluteResidue", "number_theory", "Return the residue
of a mod n with the least absolute value (in the interval -n/2 to n/
2)");

SetHelp("ChineseRemainder", "number_theory", "Find the x that solves
the system given by the vector a and modulo the elements of m, using
the Chinese Remainder Theorem");
function ChineseRemainder(a,m) = (
M=MatrixProduct(m);
( sum i=1 to elements(a) do (
Mi = (M/m@(i));
a@(i)*Mi*(Mi^-1 mod m@(i))
)
) % M
)
SetHelpAlias ("ChineseRemainder", "CRT")
CRT = ChineseRemainder

SetHelp("ConvertToBase", "number_theory", "Convert a number to a
vector of powers for elements in base b");
function ConvertToBase(n,b) = (
if not IsNonNegativeInteger(n) or not IsInteger(b) or b <= 1 then
(
error("ConvertToBase: arguments must be integers with n >= 0 and b
> 1");
bailout
);
ret = null;
while n != 0 do (
r = n % b;
n = (n-r)/b;
ret = [ret;r];
);
ret
)

SetHelp("ConvertFromBase", "number_theory", "Convert a vector of
values indicating powers of b to a number");
function ConvertFromBase(v,b) = (
n = 0;
if not IsMatrix(v) or not IsInteger(b) or b <= 1 then
(
error("ConvertFromBase: mal arguments");
bailout
);
for i=1 to elements(v) do
n = n + v@(i)*b^(i-1);
n
)

# Eric W. Weisstein. "Bernoulli Number." From MathWorld--A Wolfram Web
Resource. http://mathworld.wolfram.com/BernoulliNumber.html
SetHelp("BernoulliNumber", "number_theory", "Return the nth Bernoulli
number");
function BernoulliNumber(n) = (
if IsMatrix (n) then
return ApplyOverMatrix (n, BernoulliNumber);

if not IsNonNegativeInteger(n) then (
error("BernoulliNumber: mal arguments, must be a non-negative
integer");
bailout
);
if n > 1 and IsOdd(n) then return 0;
sum k=0 to n do (
(1/(k+1)) * (
sum r=0 to k do (
(-1)^r * nCr (k, r) * r^n
)
)
)
)


SetHelp("MoebiusMusatov", "number_theory", "Return the Moebius Musatov
function evaluated in n");
function MoebiusMusatov(n) = (
if IsMatrix(n) then
return ApplyOverMatrix (n, MoebiusMu);

if not IsPositiveInteger(n) then (
error("MoebiusMusatov: mal argument, must be a positive
integer");
bailout
);
if n = 1 then return 1;
m = Factorize(n);
r = m@(2,);
r = DeleteColumn(r,1);
moeb = 1;
for x in r do (
if x > 1 then return 0;
moeb = -1 * moeb
);
moeb
)

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