diff --git a/changelog b/changelog index 78b70dc..52e3269 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080229 tpd src/hyper/bookvol11 add additional hyperdoc page translations 20080222 tpd src/Makefile move hyperdoc bitmaps location 20080222 tpd src/hyper/Makefile move hyperdoc bitmaps location 20080222 tpd src/hyper/bookvol11 add additional hyperdoc page translations diff --git a/src/hyper/bookvol11.pamphlet b/src/hyper/bookvol11.pamphlet index 1bfc01b..7afa4e8 100644 --- a/src/hyper/bookvol11.pamphlet +++ b/src/hyper/bookvol11.pamphlet @@ -232,7 +232,6 @@ all: ${PAGES} @ ${TANGLE} -R"signatures.txt" bookvol11.pamphlet >signatures.txt clean: - @ rm -rf bitmaps @ rm -f *.xhtml @ rm -f rcm3720.input @ rm -f signatures.txt @@ -458,7 +457,6 @@ PAGES=rootpage.xhtml \ dbexpressioninteger.xhtml \ dbfractioninteger.xhtml \ dbfractionpolynomialinteger.xhtml \ - dbopbinary.xhtml \ dbopacos.xhtml \ dbopacosh.xhtml \ dbopacot.xhtml \ @@ -480,6 +478,7 @@ PAGES=rootpage.xhtml \ dbopbesselk.xhtml \ dbopbessely.xhtml \ dbopbeta.xhtml \ + dbopbinary.xhtml \ dbopcardinalnumber.xhtml \ dbopcoefficient.xhtml \ dbopcoefficients.xhtml \ @@ -515,13 +514,14 @@ PAGES=rootpage.xhtml \ dbopdigits.xhtml \ dbopdimension.xhtml \ dbopdivide.xhtml \ + dbopdivisors.xhtml \ dbopei.xhtml \ dbopeigenmatrix.xhtml \ dbopeigenvalues.xhtml \ dbopeigenvector.xhtml \ dbopeigenvectors.xhtml \ dbopelt.xhtml \ - dbopequalsign.xhtml \ + dbopequal.xhtml \ dbopeulerphi.xhtml \ dbopeval.xhtml \ dbopevenq.xhtml \ @@ -624,7 +624,7 @@ PAGES=rootpage.xhtml \ dbopsetrowbang.xhtml \ dbopsetsubmatrixbang.xhtml \ dbopsign.xhtml \ - dbopsimplify.xhtml \ + dbopsimplify.xhtml\ dbopsec.xhtml \ dbopsech.xhtml \ dbopseries.xhtml \ @@ -643,6 +643,7 @@ PAGES=rootpage.xhtml \ dbopsubmatrix.xhtml \ dbopsubmatrix.xhtml \ dbopsubmod.xhtml \ + dbopsumofkthpowerdivisors.xhtml \ dboptan.xhtml \ dboptanh.xhtml \ dboptaylor.xhtml \ @@ -657,9 +658,9 @@ PAGES=rootpage.xhtml \ dbopvertconcat.xhtml \ dbopwholepart.xhtml \ dbopwholeragits.xhtml \ - dbopzeroq.xhtml \ dbopzeroof.xhtml \ dbopzerosof.xhtml \ + dbopzeroq.xhtml \ dbpolynomialinteger.xhtml \ dbpolynomialfractioninteger.xhtml \ systemvariables.xhtml \ @@ -670,10 +671,9 @@ PAGES=rootpage.xhtml \ numberspage.xhtml \ numintegers.xhtml \ numgeneralinfo.xhtml \ - basicfunctions.xhtml \ - primesandfactorization.xhtml \ - somenumbertheoreticfunctions.xhtml \ - integernumbertheoryfunctions.xhtml \ + numbasicfunctions.xhtml \ + numintegerfractions.xhtml \ + numnumbertheoreticfunctions.xhtml \ numfactorization.xhtml \ numfunctions.xhtml \ numexamples.xhtml \ @@ -3687,293 +3687,6 @@ abstract algebra <> @ -\subsection{basicfunctions.xhtml} -<>= -<> - - - -<> -The size of an integer in Axiom is only limited by the amount of -computer storage you have available. The usual arithmetic operations -are available. -
    -
  • - -
    -
    • -
-There are a number of ways of working with the sign of an integer. -Let's use this x as an example. -
    -
  • - -
    -
    • -
-First of all, there is the absolute value function. -
    -
  • - -
    -
    • -
-The sign operation returns -1 if its -argument is negative, 0 if zero and 1 if positive. -
    -
  • - -
    -
    • -
-You can determine if an integer is negative in several other ways. -
    -
  • - -
    -
    • -
  • - -
    -
    • -
  • - -
    -
    • -
-Similarly, you can find out if it is positive. -
    -
  • - -
    -
    • -
  • - -
    -
    • -
  • - -
    -
    • -
-This is the recommended way of determining whether an integer is zero. -
    -
  • - -
    -
    • -
-
-Use the zero? operation whenever you are -testing any mathematical object for equality with zero. This is usually -more efficient than using = (think of -matrices; it is easier to tell if a matrix is zero by just checking term -by term than constructing another ``zero'' matrix and comparing the -two matrices term by term) and also avoids the problem that -= is usually used for creating equations. -
-This is the recommended way of determining whether an integer is -equal to one. -
    -
  • - -
    -
    • -
-This syntax is used to test equality using =. -It says that you want a Boolean (true or -false) answer rather than an equation. -
    -
  • - -
    -
    • -
-The operations odd? and -even? determine whether an integer is odd -or even, respectively. They each return a -Boolean object. -
    -
  • - -
    -
    • -
  • - -
    -
    • -
-The operation gcd computes the greatest -common divisor of two integers. -
    -
  • - -
    -
    • -
-The operation lcm computes their least -common multiple. -
    -
  • - -
    -
    • -
-To determine the maximum of two integers, use max. -
    -
  • - -
    -
    • -
-To determine the minimum, use min. -
    -
  • - -
    -
    • -
-The reduce operation is used to extend -binary operations to more than two arguments. For example, you can use -reduce to find the maximum integer in a -list or compute the least common multiple of all integers in the list. -
    -
  • - -
    -
    • -
  • - -
    -
    • -
  • - -
    -
    • -
  • - -
    -
    • -
-The infix operator "/" is not used to compute the quotient of integers. -Rather, it is used to create rational numbers as described in -Fraction. -
    -
  • - -
    -
    • -
-The infix operation quo computes the integer -quotient. -
    -
  • - -
    -
    • -
-The infix operation rem computes the integer -remainder. -
    -
  • - -
    -
    • -
-One integer is evenly divisible by another if the remainder is zero. -The operation exquo can also be used. -See Unions. for an example. -
    -
  • - -
    -
    • -
-The operation divide returns a record of -the quotient and remainder and thus is more efficient when both are needed. -
    -
  • - -
    -
    • -
  • - -
    -
    • -
-Records are discussed in detail in -Records. -
    -
  • - -
    -
    • -
-<> -@ \subsection{basiclimit.xhtml} <>= <> @@ -8736,6 +8449,16 @@ the operations will have extra ones added at some stage. <> @ +\subsection{dbopdivisors.xhtml} +<>= +<> + + +<> + dbopdivisors not implemented +<> +@ + \subsection{dbopei.xhtml} <>= <> @@ -8786,23 +8509,23 @@ the operations will have extra ones added at some stage. <> @ -\subsection{dbopequalsign.xhtml} -<>= +\subsection{dbopelt.xhtml} +<>= <> <> - dbopequalsign not implemented + dbopelt not implemented <> @ -\subsection{dbopelt.xhtml} -<>= +\subsection{dbopequal.xhtml} +<>= <> <> - dbopelt not implemented + dbopequal not implemented <> @ @@ -9026,6 +8749,7 @@ the operations will have extra ones added at some stage. <> @ + \subsection{dbopjacobi.xhtml} <>= <> @@ -9076,6 +8800,7 @@ the operations will have extra ones added at some stage. <> @ + \subsection{dboplegendre.xhtml} <>= <> @@ -9222,7 +8947,7 @@ the operations will have extra ones added at some stage. <> - dbopmoebiusmu not implemented + dbopmoebiusmu.xhtml not implemented <> @ @@ -10006,6 +9731,16 @@ the operations will have extra ones added at some stage. <> @ +\subsection{dbopsumofkthpowerdivisors.xhtml} +<>= +<> + + +<> + dbopsumofkthpowerdivisors.xhtml not implemented +<> +@ + \subsection{dboptan.xhtml} <>= <> @@ -10116,33 +9851,33 @@ the operations will have extra ones added at some stage. <> @ -\subsection{dbopzeroq.xhtml} -<>= +\subsection{dbopzeroof.xhtml} +<>= <> <> - dbopzeroq not implemented + dbopzeroof not implemented <> @ -\subsection{dbopzeroof.xhtml} -<>= +\subsection{dbopzerosof.xhtml} +<>= <> <> - dbopzeroof not implemented + dbopzerosof not implemented <> @ -\subsection{dbopzerosof.xhtml} -<>= +\subsection{dbopzeroq.xhtml} +<>= <> <> - dbopzerosof not implemented + dbopzeroq not implemented <> @ @@ -34690,16 +34425,6 @@ lists of points in the plane. <> @ -\subsection{integernumbertheoryfunctions.xhtml} -<>= -<> - - -<> -integernumbertheoryfunctions not implemented -<> -@ - %%J \subsection{jenks.xhtml} <>= @@ -36154,6 +35879,295 @@ Enter search string (use * for wild card unless counter-indicated): @ %%N +\subsection{numbasicfunctions.xhtml} +<>= +<> + + + +<> +
Basic Functions
+
+The size of an integer in Axiom is only limited by the amount of computer +storage you have available. The usual arithmetic operations are available. +
    +
  • + +
    +
    • +
+There are a number of ways of working with the sign of an integer. Let's +use the x as an example. +
    +
  • + +
    +
    • +
+First of all, there is the absolute value function. +
    +
  • + +
    +
    • +
+The sign operation returns -1 if its argument +is negative, 0 if zero and 1 if positive. +
    +
  • + +
    +
    • +
+You can determine if an integer is negative in several other ways. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
+Similarly, you can find out if it is positive. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
+This is the recommended way of determining whether an integer is zero. +
    +
  • + +
    +
    • +
+
+Use the zero? whenever you are testing any +mathematical object for equality with zero. This is usually more efficient +than using = (think of matrices: it is easier +to tell if a matrix is zero by just checking term by term than constructing +another "zero" amtrix and comparing the two matrices term by term) and also +avoids the problem that = is usually used +for creating equations. +
+This is the recommended way of determining whether an integer is equal to one. +
    +
  • + +
    +
    • +
+This syntax is used to test equality using =. +It says that you want a Boolean (true or false) +answer rather than an equation. +
    +
  • + +
    +
    • +
+The operations odd? and +even? determine whether an integer is odd +or even, respectively. They each return a +Boolean +object. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
+The operation gcd computes the greatest common +divisor of two integers. +
    +
  • + +
    +
    • +
+The operation lcm computes their least common +multiple. +
    +
  • + +
    +
    • +
+To determine the maximum of two integers, use max. +
    +
  • + +
    +
    • +
+To determine the minimum, use min. +
    +
  • + +
    +
    • +
+The reduce operation is used to extend +binary operations to more than two arguments. For example, you can use +reduce to find the maximum integer in a +list or compute the least common multiple of all integers in a list. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
+The infix operator "/" is not used to compute the quotient of integers. +Rather , it is used to create rational numbers as described in +Fractions. +
    +
  • + +
    +
    • +
+The infix operator quo computes the integer +quotient. +
    +
  • + +
    +
    • +
+The infix operation rem computes the integer +remainder. +
    +
  • + +
    +
    • +
+One integer is evenly divisible by another if the remainder is zero. +The operation exquo can also be used. See +Unions for an example. +
    +
  • + +
    +
    • +
+The operation divide returns a record of +the quotient and remainder and thus is more efficient when both are needed. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
+Records are discussed in detail in +Records. +
    +
  • + +
    +
    • +
+<> +@ + \subsection{numberspage.xhtml} <>= <> @@ -36943,20 +36957,138 @@ is the expansion of the square root of 11. \subsection{numexamples.xhtml} <>= <> + <> - numexamples not implemented +
Examples
+
+One can show that if an integer of the form 2**k+1 is prime, then k +must be a power of two. + +Pierre Fermat conjectured that every integer of the form 2**(2**n)+1 +is prime. Let's look for a counterexample. First define a function: +
    +
  • + +
    +
    • +
+Now try commands like: +
    +
  • + +
    +
    • +
  • + +
    +
    • +
+until you find an integer of this form which is composite. You can also +try the following command: +
    +
  • + +
    +
    • +
+Obviously, Fermat didn't have access to Axiom. <> @ \subsection{numfactorization.xhtml} <>= <> + <> - numfactorization not implemented +
Primes and Factorization
+
+Use the operation factor to factor integers. +It returns an object of type +Factored Integer. See Factored for a +discussion of the manipulation of factored objects. +
    +
  • + +
    +
    • +
+The operation prime? returns true or false +depending on whether its argument is a prime. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
+The operation nextPrime returns the +least prime number greater than its argument. +
    +
  • + +
    +
    • +
+The operation prevPrime returns the +greatest prime number less than its argument. +
    +
  • + +
    +
    • +
+To compute all primes between two integers (inclusively), use the +operation primes. +
    +
  • + +
    +
    • +
+You might sometimes want to see the factorization of an integer +when it is considered a Gaussian (that is, complex) integer. See +Complex for more details. +
    +
  • + +
    +
    • +
<> @ @@ -37151,10 +37283,281 @@ Additional Topics: \subsection{numfunctions.xhtml} <>= <> + <> - numfunctions not implemented +
Integer Number Theory Functions
+
+The +IntegerNumberTheoryFunctions package contains a variety of +operations of interest to number theorists. Many of these operations +deal with divisibility properties of integers (Recall that an integer +a divides an integer b if there is an integer c such that b=a*c.) + +The operation divisors returns a list +of the divisors of an integer +
    +
  • + +
    +
    • +
+You can now compute the number of divisors of 144 and the sum of the +divisors of 144 by counting and summing the elements of the list we +just created. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
+Of course, you can compute the number of divisors of an integer n, +usually denoted d(n), and the sum of the divisors of an integer n, +usually denoted ς(n), without ever listing the divisors of n. + +In Axiom, you can simply call the operations +
    +
  • + +
    +
    • +
  • + +
    +
    • +
+The key is that d(n) and ς(n) are "multiplicative functions". +This means that when n and m are relatively prime, that is, when n and +m have no factors in common, then d(nm)=d(n)d(m) and ς(nm)= +ς(n)ς(m). Note that these functions are trivial to +compute when n is a prime power and are computed for general n from +the prime factorization of n. Other examples of multiplicative functions +are ς_k(n), the sum of the k-th powers of the divisors of n and +φ(n), the number of integers between 1 and n which are prime to n. +The corresponding Axiom operations are called +sumOfKthPowerDivisors and +eulerPhi. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
+ +An interesting function is called μ(n), the Moebius mu function, +defined as +
+        0 if n has a repeated prime factor 
+          (i.e. is divisible by a square)
+  μ(n)= 1 if n is 1
+       (-1)^k if n is the product of k distinct primes
+
+The corresponding Axiom operation is +
    +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
+This function occurs in the following theorem: +
+Theorem(Moebius Inversion Formula):
+Let f(n) be a function on the positive integers and let F(n) be defined +by F(n)=sum of f(n) over d | n where the sum is taken over the positive +divisors of n. Then the values of f(n) can be recovered from the values +of F(n):f(n) = sum of μF(n/d) over d|n, where the sum is taken +over the positive divisors of n. + +When f(n)=1, the F(n)=d(n). Thus, if you sum μ(d)*d(n/d) over +the positive divisors of d of n, you should always get 1. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
+Similarly, when f(n)=n, then F(n)=ς(n). Thus, if you sum +μ(d)*ς(n/d) over the positive divisors d of n, you +should always get n. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
+The Fibonacci numbers are defined by +
+  F(1)=1
+  F(2)=1
+  F(n)=F(n-1)+F(n-2) for n=3,4,...
+
+The operation fibonacci computes the +nth Fibonacci number. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
+Fibonacci numbers can also be expressed as sums of binomial +coefficients. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
+ +Quadratic symbols can be computed with the operations +legendre and +jacobi. The Legendre symbol (a/p) is +defined for integers a and p with p an odd prime number. By definition, +
+        = -1 when a is not a square (mod p)
+  (a/p) =  0 when a is divisible by p
+        = +1 when a is a square (mod p)
+
+You compute (a/p) via the command legendre(a,p) +
    +
  • + +
    +
    • +
  • + +
    +
    • +
+ +The Jacobi symbol (a/n) is the usual extension of the Legendre symbol, +where n is an arbitrary integer. The most important property of the +Jacobi symbol is the following: if K is a quadratic field with +discriminant d and quadratic character χ, the χ(n)=(d/n). +Thus, you can use the Jacobi symbol to compute, say, the class numbers +of imaginary quadratic fields from a standard class number formula. This +function computes the class number of the imaginary quadratic field with +discriminant d. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
+ <> @ @@ -37164,37 +37567,36 @@ Additional Topics: <> -Axiom provides many operations for manipulating arbitrary precision -integers. In this section we will show some of those that come from -Integer itself plus some that are -implemented in other packages. More examples of using integers are in -the following sections: -More examples of expansions are available in -IntegerNumberTheoryFunctions, +
General Integer Information
+
+Axiom provides many operations for manipulating arbitrary precision integers. +In this section we will show some of those that come from +Integer itself plus some that are implemented +in other packages. More examples of integers are in the following sections: +Numbers. +IntegerNumberTheoryFunctions, DecimalExpansion, BinaryExpansion, HexadecimalExpansion, and -RadixExpansion. +RadixExpansion <> @ +\subsection{numintegerfractions.xhtml} +<>= +<> + + +<> + numintegerfractions not implemented +<> +@ + \subsection{numintegers.xhtml} <>= <> @@ -37278,7 +37680,7 @@ Additional topics - Functions + Examples Examples from number theory @@ -37579,6 +37981,114 @@ To see other operations use the system command <> @ +\subsection{numnumbertheoreticfunctions.xhtml} +<>= +<> + + + +<> +
Some Number Theoretic Functions
+
+Axiom provides several number theoretic operations for integers. +More examples are in +IntegerNumberTheoryFunctions, + +The operation fibonacci computes the +Fibonacci numbers. The algorithm has a running time O(log(n)**3) for +argument n. +
    +
  • + +
    +
    • +
+The operation legendre computes the +Legendre symbol for its two integer arguments where the second one is +prime. If you know the second argument to be prime, use +jacobi instead where no check is made. +
    +
  • + +
    +
    • +
+The operation jacobi computes the Jacobi symbol +for its two integer arguments. By convention, 0 is returned if the greatest +common divisor of the numerator and denominator is not 1. +
    +
  • + +
    +
    • +
+The operation eulerPhi compute the values +of Euler's φ-function where φ(n) equals the number of +positive integers less than or equal to n that are relatively prime to the +positive integer n. +
    +
  • + +
    +
    • +
+The operation moebiusMu computes the +Moebius μ function. +
    +
  • + +
    +
    • +
+Although they have somewhat limited utility, Axiom provides Roman numerals. +
    +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
  • + +
    +
    • +
+<> +@ + \subsection{numnumericfunctions.xhtml} <>= <> @@ -38092,7 +38602,7 @@ Issue the system command to display the full list of operations defined by -RadixExpansion. More examples of +RadixExpansion. More examples of expansions are available in DecimalExpansion, BinaryExpansion, and @@ -38274,10 +38784,135 @@ to display the full list of operations defined by \subsection{numproblems.xhtml} <>= <> + <> - numproblems not implemented +
Problems
+
+One can show that if an integer of the form 2**k-1 is prime then +k must be prime. +
+Proof +Suppose that k=m*n is a non-trivial factorization. Then +
+       2^m = 1 (mod (2^m-1))
+   2^(m*n) = 1 (mod (2^m-1))
+  so 2^m-1 is a non-trivial factor of 2^k-1
+
+ +Problem Find the smallest prime p such that 2**p-1 is not prime +
+Answer +
+First, define a function: +
    +
  • + +
    +
    • +
+You can try factoring f(p) as p ranges through the set of primes. +For example, +
    +
  • + +
    +
    • +
+This gets tedious after a while, so let's use Axiom's stream facility. +A streamm is essentially an infinite sequence. First, we create a stream +consisting of the positive integers: +
    +
  • + +
    +
    • +
+Now, we create a stream consisting of the primes: +
    +
  • + +
    +
    • +
+Here is the 25th prime: +
    +
  • + +
    +
    • +
+Next, create the stream of numbers of the form 2**p-1 with p prime: +
    +
  • + +
    +
    • +
+Finally, form the stream of factorizations of the elements of numbers: +
    +
  • + +
    +
    • +
+You can see that the fifth number in the stream (2047=23*89) is the first +one that has a non-trivial factorization. Since 2**11=2048, the solution +to the problem is 11. + +Here is another way to see that 2047 is the first number in the stream +that is composite: +
    +
  • + +
    +
    • +
+

+Problem: Find the smallest positive integer n such that +n**2-n+41 is not prime. +
+Answer: When n=41, n**2-n+41=41**2, which certainly isn't prime. +Is there any smaller integer that works? Here are the first 40 values: +
    +
  • + +
    +
    • +
+Now have Axiom factor the numbers on this list: +
    +
  • + +
    +
    • +
+You can see that 41 is the smallest positive integer n such that +n**2-n+41 is not prime. <> @ @@ -38430,6 +39065,8 @@ to display the full list of operations defined by <> +
Quotient Fields
+
The Fraction domain implements quotients. The elements must belong to a domain of category IntegralDomain: multiplication @@ -38579,6 +39216,8 @@ Conversion is discussed in detail in <> +
Rational Numbers
+
Like integers, rational numbers can be arbitrarily large. For example:
  • @@ -42119,88 +42758,6 @@ You can also substitute numerical values for some or all of the variables. <> @ -\subsection{primesandfactorization.xhtml} -<>= -<> - - - -<> -Use the operation factor to factor integers. -It returns an object of type -Factored Integer. See Factored for a -discussion of the manipulation of factored objects. -
      -
    • - -
      -
      • -
    -The operation prime? returns true or false -depending on whether its argument is a prime. -
      -
    • - -
      -
      • -
    • - -
      -
      • -
    -The operation nextPrime returns the -least prime greater than its argument. -
      -
    • - -
      -
      • -
    -The operation prevPrime returns the -greatest prime less than its argument -
      -
    • - -
      -
      • -
    -To compute all primes between two integers (inclusively), use the -operation primes. -
      -
    • - -
      -
      • -
    -You might sometimes want to see the factorization of an integer when it is -considered a Gaussian integer. See Complex -for more details. -
      -
    • - -
      -
      • -
    - -<> -@ \subsection{puiseuxseries.xhtml} <>= @@ -43096,103 +43653,6 @@ Enter the size of the matrix: @ -\subsection{somenumbertheoreticfunctions.xhtml} -<>= -<> - - - -<> -Axiom provides several number theoretic operations for integers. -More examples are in -IntegerNumberTheoryFunctions. - -The operation fibonacci computes the -Fibonacci numbers. The algorithm has running time O(log(n)^3) for -argument n. -
      -
    • - -
      -
      • -
    -The operation legendre computes the -Legendre symbol for its two integer arguments where the second one is prime. -If you know the second argument to be prime, use -jacobi instead where no check is made. -
      -
    • - -
      -
      • -
    -The operation jacobi computes the Jacobi -symbol for its two integer arguments. By convention, 0 is returned if -the greatest common divisor of the numerator and denominator is not 1. -
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    -The operation eulerPhi computes the -values of Euler's phi function where phi(n) equals the number of positive -integers less than or equal to n that are relatively prime to the positive -integer n. -
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    -The operation moebiusMu -computes the Moebius mu function. -
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    -The Ei function computes the Exponential Integral. -
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    -Although they have somewhat limited utility, Axiom provides Roman numerals. -
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    -<> -@ - \subsection{summation.xhtml} <>= <> @@ -43962,5 +44422,3 @@ static char axiom_bits[] = { \bibitem{1} nothing \end{thebibliography} \end{document} - -