3.2.4 Integer and Number Theoretical Functions

Transcription

1 Advaced Mathematics i Mathematica Iteger ad Number Theoretical Fuctios Mod[k, ] k modulo (positive remaider from dividig k by ) Quotiet[m, ] the quotiet of m ad (iteger part of m/) GCD[ 1, 2, :::] the greatest commo divisor of 1, 2,... LCM[ 1, 2, :::] the least commo multiple of 1, 2,... Some iteger fuctios. The remaider o dividig 17 by 3. I[1]:= Mod[17, 3] Out[1]= 2 The iteger part of 17=3. I[2]:= Quotiet[17, 3] Out[2]= 5 Mod also works with real umbers. I[3]:= Mod[5.6, 1.2] Out[3]= 0.8 Mod always gives a o-egative result. I[4]:= Mod[-5.6, 1.2] Out[4]= 0.4 For ay itegers a ad b,it is always true that b*quotiet[a, b] + Mod[a, b] is equal to a. The greatest commo divisor fuctio GCD[ 1, 2, :::] gives the largest iteger that divides all the i exactly. Whe you eter a ratio of two itegers, Mathematica eectively uses GCD to cacel out commo factors, ad give a ratioal umber i lowest terms. The least commo multiple fuctio LCM[ 1, 2, :::] gives the smallest iteger that cotais all the factors of each ofthe i. The largest iteger that divides both 24 ad 15 is 3. I[5]:= GCD[24, 15] Out[5]= 3

2 3.2 Mathematical Fuctios 419 FactorIteger[] a list of the prime factors of, ad their expoets Divisors[] a list of the itegers that divide Prime[k] the k th prime umber PrimeQ[] give True if is a prime, ad False otherwise Iteger factorig ad related fuctios. This gives the factors of 24 as 2 3,3 1. The rst elemet ieach list is the factor the secod is its expoet. Here are the factors of a larger iteger. I[6]:= FactorIteger[24] Out[6]= {{2, 3}, {3, 1}} I[7]:= FactorIteger[ ] Out[7]= {{3, 2}, {7, 1}, {11, 1}, {13, 1}, {19, 1}, {37, 1}, {52579, 1}, {333667, 1}} You should realize that accordig to curret mathematical thikig, iteger factorig is a fudametally dicult computatioal problem. As a result, you ca easily type i a iteger that Mathematica will ot be able to factor i aythig short of a astroomical amout of time. So log as the itegers you give are less tha about 20 digits log, FactorIteger should have o trouble. Oly i special cases, however, will it be able to deal with much loger itegers. (You ca make some factorig problems go faster by settig the optio FactorComplete->False, so that FactorIteger[] tries to pull out oly oe factor from.) Here is a rather special log iteger. I[8]:= 30! Out[8]= Mathematica ca easily factor this special iteger. I[9]:= FactorIteger[%] Out[9]= {{2, 26}, {3, 14}, {5, 7}, {7, 4}, {11, 2}, {13, 2}, {17, 1}, {19, 1}, {23, 1}, {29, 1}} Although Mathematica may ot be able to factor a large iteger, it ca ofte still test whether or ot the iteger is a prime. I additio, Mathematica has a fast way to d the k th prime umber. It is ofte much faster to test whether a umber is prime tha to factor it. I[10]:= PrimeQ[ ] Out[10]= False

3 Advaced Mathematics i Mathematica Here is a plot of the rst 100 primes. I[11]:= ListPlot[ Table[ Prime[], {, 100} ] ] This is the millioth prime. I[12]:= Prime[ ] Out[12]= PowerMod[a, b, ] EulerPhi[] MoebiusMu[] DivisorSigma[k, ] JacobiSymbol[, m] ExtededGCD[m, ] LatticeReduce[{v 1, v 2, :::}] the power a b modulo the Euler totiet fuctio () the Mobius fuctio () the divisor fuctio k () the Jacobi symbol ; m the exteded gcd of m ad the reduced lattice basis for the set of iteger vectors v i Some fuctios from umber theory. The modular power fuctio PowerMod[a, b, ] gives exactly the same results as Mod[a^b, ]. PowerMod is much more eciet, however, because it avoids geeratig the full form of a^b. You ca use PowerMod ot oly to d positive modular powers, but also to d modular iverses. PowerMod[a, -b, ] gives, if possible, a iteger k such that (ak) b 1mod. (Wheever such aiteger exists, it is guarateed to be uique.) If o such iteger k exists, Mathematica leaves PowerMod uevaluated. PowerMod is equivalet to usig Power, the Mod, but is much more eciet. I[13]:= PowerMod[2, 13451, 3] Out[13]= 2 This gives the modular iverse of 3 modulo 7. I[14]:= PowerMod[3, -1, 7] Out[14]= 5

4 3.2 Mathematical Fuctios 421 Multiplyig the iverse by 3 modulo 7 gives 1, as expected. I[15]:= Mod[3 %, 7] Out[15]= 1 The Euler totiet fuctio () gives the umber of itegers less tha that are relatively prime to. A importat relatio (Fermat's Little Theorem) is that a () 1mod for all a relatively prime to. The Möbius fuctio () is deed to be (;1) k if is a product of k distict primes, ad 0 if cotais a squared factor (other tha 1). A importat relatio is the Mobius iversio formula, which states that if g() = P dj f(d), the f() = (d)g(=d), where the sums are over all positive itegers d that divide. P dj The divisor fuctio k () isthesumofthek th powers of the divisors of. The fuctio 0 () gives the total umber of divisors of, ad is ofte deoted d(). The fuctio 1 (), equal to the sum of the divisors of, ad is ofte deoted (). For prime, () = ; 1. I[16]:= EulerPhi[17] Out[16]= 16 The result is 1, as guarateed by Fermat's Little Theorem. I[17]:= PowerMod[3, %, 17] Out[17]= 1 This gives a list of all the divisors of 24. I[18]:= Divisors[24] Out[18]= {1, 2, 3, 4, 6, 8, 12, 24} 0() gives the total umber of distict divisors of 24. I[19]:= DivisorSigma[0, 24] Out[19]= 8 The Jacobi symbol JacobiSymbol[, m] reduces to the ; Legedre symbol m whe m is a odd prime. The Legedre symbol is equal to zero if is divisible by m, otherwise it is equal to 1 if is a quadratic residue modulo the prime m, ad to ;1 if it is ot. A iteger relatively prime to m is said to be a quadratic residue modulo m if there exists a iteger k such that k 2 mod m. The full Jacobi symbol is a product of the Legedre symbols p i for each of the prime factors p i such that m = Q i p i. The exteded gcd ExtededGCD[m, ] gives a list{g, {r, s}} where g is the greatest commo divisor of m ad, ad r ad s are itegers such that g = rm+s. The exteded gcd is importat i dig iteger solutios to liear (Diophatie) equatios.

5 Advaced Mathematics i Mathematica The rst umber i the list is the gcd of 105 ad 196. I[20]:= ExtededGCD[105, 196] Out[20]= {7, {15, -8}} The secod pair of umbers satises g = rm + s. I[21]:= Out[21]= 7 The lattice reductio fuctio LatticeReduce[{v 1, v 2, :::}] is used i may moder umber theoretical ad combiatorial algorithms. The basic idea is to thik of the vectors v k of itegers as deig a mathematical lattice. The vector represetig each poit i the lattice ca be writte as a liear combiatio of the form P c k v k, where the c k are itegers. For a particular lattice, there are may possible choices of the \basis vectors" v k. What LatticeReduce does is to d a reduced set of basis vectors v k for the lattice, with certai special properties. Three uit vectors alog the three coordiate axes already form a reduced basis. This gives the reduced basis for a lattice i four-dimesioal space specied by three vectors. I[22]:= LatticeReduce[{{1,0,0},{0,1,0},{0,0,1}}] Out[22]= {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}} I[23]:= LatticeReduce[{{1,0,0,12345}, {0,1,0,12435}, {0,0,1,12354}}] Out[23]= {{-1, 0, 1, 9}, {9, 1, -10, 0}, {85, -143, 59, 6}} Notice that i the last example, LatticeReduce replaces vectors that are early parallel by vectors that are more perpedicular. I the process, it ds some quite short basis vectors.