2. The binary number system and the binomial coefficients

Transcription

1 . The biary umber system ad the biomial coefficiets Before we proceed with the itroductio of the stepumber system let us review the biary umber system i this Chapter ad Cator s, usig ifiitely may digits, i the ext. I eumeratig biary umbers we are doig more tha simply puttig the atural umbers ito biary code. We are also coutig the umber of fiite subsets, as well as costructig them. For example, if we wat to cout the umber of subsets of a set of four elemets, we tur to the up-frot-0 represetatio of biary umbers: 0000, 000, 000, 00, 000, 00, 00, 0, 000, 00, 00, 0, 00, 0, 0, Not oly have we couted the umber of subsets of a set of four elemets (ad foud it to be 6) but we have also costructed every oe of them, e.g., 0000 is the trivial subset ad is the uiversal subset. The costructio uses the idea of a characteristic fuctio ad the covetio that the places which the digits occupy costitute the elemets of a ordered set. I our example that set has four elemets, amely, the four places we eed to write a 4-digit umber. A characteristic fuctio is defied by postulatig that the digit idicates that the place i appositio is icluded i the subset; the digit 0 idicates that it is ot. Thus 00 represets the two-elemet subset cosistig of the secod ad fourth elemets of the 4-set. I this way every subset of the 4-set has bee accouted for oce, ad oly oce. It is clear that we ca also set up a oe-to-oe correspodece betwee the subsets of a -set ad biary umbers of at most digits for all. I view of the foregoig discussio o represetig subsets by biary umbers, we ca give a equivalet defiitio of the biomial coefficiets as follows. Give ad, 0, is the umber of biary umbers of at most digits of which exactly are valuable. We proceed to review the basic properties of the biomial coefficiets here i order to establish the laguage ad patter that we shall be usig i treatig factorial coefficiets i relatio to Cator s umber system, ad spectral coefficiets i relatio to the stepumber system, as well as Stirlig umbers of either id. The equivalece of the two defiitios of the biomial coefficiets is a simple cosequece of the fact that coutig subsets ca be doe by coutig characteristic fuctios, which i tur ca be doe by coutig biary umbers i up-frot-0 represetatio. It is immediate that 0 = ad =, as there is oly oe biary umber with o valuable digits, amely 0, ad there is oly oe of exactly digits all of which are valuable, amely.we also have the Symmetry Property = 9

2 This property has o couterpart for factorial coefficiets of Cator s umber system, for spectral coefficiets of the stepumber system, or for Stirlig umbers. To prove it we may observe that, give a biary umber a of valuable digits, the equatio x + a = has a uique solutio which, as a biary umber, has exactly valuable digits. Recursio Formula = + It is, i fact, a differece equatio which, uder the Iitial Coditio = yields a uique solutio, coveietly tabulated i the form proposed by Blaise Pascal (63-66), with stadig i the th place of the th row (the coutig of places ad rows starts with 0). Biomial coefficiets i Pascal's triagle A more extesive table ca be foud at the ed of the boo. Pascal s iovatio i tabulatig the biomial coefficiets i this format has become the source of a wealth of iformatio. Summatio Formula 0

3 = The familiar idetity ( ) + = + furishes the Overflow Formula + = 0 To prove the Recursio Formula let us distiguish oe place by callig it zero place, ad survey the biary umbers accordig as they have the digit 0 or the digit i the zero place. I this maer we have cosidered every biary umber of at most digits of which exactly are valuable oce ad oly oce. This completes the proof. The biomial coefficiets ear their ame by the role they play i the Biomial Theorem + x = + x x x ( )... Sierpisi s Triagles (mod p) If we blot out the etries of Pascal s triagle ad replace them with dummies,,,,, accordig as the etry i questio is cogruet to 0,,, 3,..., p (mod p), the we get what is ow as the (colored) Sierpisi triagle. Lie Pascal s, it is ifiite. Let p be a odd prime umber. The biomial coefficiets i row p betwee etries ad p iclusive, i row p + betwee etries ad p iclusive, i row p + 3 betwee etries 3 ad p 3 iclusive, etc.,..., are divisible by p, for =,, 3,... The patter cotiues all the way dow to row p wherei the (p ) st p etry is divisible by p. This is the low vertex of a iverted p equilateral triagle of height p cosistig of etries cogruet to 0 (mod p). Further scrutiy reveals that they occur i other places as well throughout Pascal s triagle. Each iverted triagle is uiquely determied by its apex at 3p 3p 4p 4p 4p 5p 5p 5p 5p, ;,, ;,,, ; p p p p 3p p p 3p 4p ( =,, 3,...), maig up a itriguig, repetitive, fractal patter that was discovered by the Polish mathematicia Waclaw Sierpisi (88-969). The fractal homothety ratios are: p, p, p 3..., p,... with ceter at the apex of Pascal s triagle. It is importat to realize that this is true for prime umbers oly. It fails for composite umbers, e.g., there are odd umbers i row 6 other tha etry 0 ad 6 (mod 6), divisible by 6. Sierpisi s Triagle for Biomial Coefficiets (mod )

4 Code: 0, (mod )

5 Sierpisi s Triagle for Biomial Coefficiets (mod 3) Code: 0,, (mod 3) Sierpisi s Triagle for Biomial Coefficiets (mod 5) 3

6 Code: 0,,, 3, 4 (mod 5) Biomial coefficiets i Sierpisi s triagle (mod 7) Code: 0,,, 3, 4, 5, 6 (mod 7) 4

7 x

8 The Geeralized Little Fermat Theorem p The Little Fermat Theorem states that a a (mod p) where p is prime. If, i additio, p does ot p divide a the by the Cacellatio Law we also have a (mod p). The proof is by mathematical iductio. For a = there is a proof without words. Behold Sierpisi s triagle (mod p) ad see that there are oly two o-zero dummies i row p, the first ad the last, each equal to. O the other had, by the Summatio Formula, the sum of etries i the p th row is p. We coclude that p (mod p). To exted this to a we eed the mod p versio of the Biomial Formula: ( a+ b) p a p + b p (mod p) that ca also be proved without words, just by beholdig Sierpisi s triagle. Now we ca p p p p p p start the iductio: 3 = ( + ) + + = 3 (mod p); 4 = (3 + ) = 4 (mod p); etc., ad the result follows. p This is just a special case, for =, of a more geeral result: a (mod p), =,, 3, The proof of the full stregth of the Little Fermat Theorem is left as a exercise. There are may other proofs of the type behold! They are all relegated to the Exercises. Newto s First ad Secod Formula It is curious that authors, amog them pioeers such as the America mathematicia Eric Temple Bell, were blid to the fact that the Biomial Theorem is valid ot oly for umbers but operators as well. Their blidess might have bee due to a otatioal stumblig bloc: they had thought that they were obliged to ivet a ew symbol (e.g., I or ι) for the idetity operator, bypassig the obvious choice of. I this treatise we shall be cocered with several operators that apply to sequeces, some of which we itroduce right ow. The shift operator E is defied such that for every sequece a we have Ea = a + (E shifts the members of the sequece by oe slot to the right). The powers of the shift operator E, E 3,... called higher order shift operators will also be used. They mae a right shift by, 3,... slots. The differece operator Δ is defied such that Δa = a + a, ad the idetity operator such that a = a for every sequece a. The fact that has aother meaig also, that of the umber oe, is ot disturbig. The powers of Δ: Δ, Δ 3,... are called higher order differece operators. For example it is easy to show that, for the geometric progressio a =, we have Δ = for =,, 3,... Ideed, Δ = + = ( ) = (for this reaso the sequece plays the same role with respect to the differece operator Δ as the expoetial fuctio e x does with respect to the differetial operator d/dx). Oe ca also cosider the iverse shift operator E defied by the formula E (a ) = a where a is a sequece with values for =...,, 0,,,... It satisfies E E = = EE, where is the idetity operator. The ext two results are ot usually treated i the cotext of the Biomial Theorem, eve though that is where they properly belog. Correctly uderstood, they are ot a formula but a algorithm. E Newto s first formula = +... Δ+ Δ + + Δ Newto s secod formula 6

9 Δ = E E E... ( ) For example we calculate E via Newto s first formula (eve though maual calculatio is quicer). E = ( + Δ) = [ + Δ + Δ Δ ] 0. = [ ] = ( + ) = +. As aother example, we re-calculate Δ via Newto s secod: Δ = (E ) = [E E ( ) ] 0. = [ ( ) ] = ( ) = for =,, 3,... Newto s formulas ca be proved by observig that E = + Δ, Δ = E ad expadig ( + Δ), (E ) via the biomial formula uder the covetio that Δ 0 = E 0 =, the idetity operator. We ca calculate the th term of a arbitrary sequece via Newto s first. For example, let us cotiue the etries i the slatig row a 0 =, 4, 0, 0, 35,..., a,... of Pascal s triagle ad calculate 7 th term. The higher differeces are: The third differece sequece is costat, so the fourth ad each subsequet higher differece is the zero sequece. Hece i Newto s first formula we have to write the first four terms oly: a 7 = E 7 a 0 = ( + 7Δ + Δ + 35Δ 3 )a 0 = + 7(3) + (3) + 35() = 0. To chec, we cotiue the sequece a maually usig the fact that Δ 3 a = is costat. The sums 6 + = 7, + 7 = 8, = 84 = a 6 we eter i the 6 th slatig row of slope +, ad cotiue to a 7 : Ours is a example of a arithmetic progressio of order 3. More geerally, members of a sequece a are said to be i arithmetic progressio of order if the th differece sequece is costat 0 (hece, all higher differece sequeces are equal to the zero sequece). A ordiary arithmetic progressio is of the first order. Pascal s triagle furishes a example of a th order arithmetic progressios for every, amely, etries i the th slatig row. Furthermore, earlier slatig rows are exactly the higher order differece sequeces, revealig that the th differece sequece is costat equal to. I fact, Pascal s triagle ca be used as a ready recoer for the higher differeces of biomial coefficiets stadig i a slatig row. They are the biomial coefficiets stadig i earlier slatig rows. I fact, rotatio of Pascal s triagle i the couter-clocwise sese through 35 will put these rows i the customary horizotal positio. A most importat arithmetic progressios of order is the sequece of the th powers: 0,,, 3,...,,... The th differece sequece is costat ad is equal to! 7

10 It is ot hard to prove the theorem that members of a sequece a are i arithmetic progressio of order if, ad oly if, a is a polyomial of degree i the variable. As a applicatio of Newto s formulas we shall prove the followig Theorem. The solutio of the system of liear equatios with ifiitely may uows x 0, x, x,... x 0 = y 0 x 0 + x = y x 0 + x + x = y x 0 + 3x + 3x + x 3 = y where o the left-had side we have the biomial coefficiets, ad y 0, y, y,... are arbitrary costats, is: x 0 = y 0 x = y 0 + y x = y 0 y + y x 3 = y 0 + 3y 3y + y where o the right-had sides we have the biomial coefficiets with alteratig sigature. Proof. By Newto s first we may write the proposed solutio i the form: x = E x 0 = (E ) y 0 E x 0 = Δ y 0 for = 0,,, 3,... Therefore (E + ) x 0 = E x 0 + E - x x 0 + E = Δ y 0 + Δ - y Δy = (Δ + ) y 0 = E y 0 = y by Newto s secod. It follows that the values x = (E ) y 0 satisfy the system, completig the proof. The coverse is also true: if the roles of the uows ad costats are iterchaged, the solutio to the secod system for the uows y is furished i terms of the costats x by the first. As a example, cosider the system of liear equatios with ifiitely may uows x = 0 x + x = 8 x 0 y 0

11 x + x + x 3 = x + 3x + 3x 3 + x 4 = The solutio is: x = (0 ) x = ( ) (0 ) x 3 = ( ) ( ) + (0 ) x 4 = (3 ) 3( ) + 3( ) (0 ) I order to iterpret this result we ote that 0,,, 3,... are i arithmetic progressio of order ad Newto s secod formula applies: Δ 0 = (E ) 0 = [E E - + E ( ) ] 0 We coclude that the solutio to the above ifiite system of liear equatios is: The umbers Δ 0 x = Δ 0 = ( ) + ( ) ( ) - have a importat (albeit little recogized) iterpretatio: Implicit formula for the umber of surjective fuctios Surj(, ) = Δ 0 where Surj(, ) stads for the umber of surjective fuctios from a -set to a -set (alteratively, the umber of distributios of ulie balls ito ulie cells with o cell to remai empty). Agai, this is ot so much a formula tha a algorithm. (The properties of ijective, surjective, ad bijective fuctios will be discussed i Volume II, Chapter 5 o duality.) For example, let = 3 ad calculate Surj(3, ). Δ 4 3 = (E ) 4 3 = (E E + ) 4 3 = E 4 3 E = 6 3 (5 3 ) = 6 (5) + 64 = = 30 = Surj(3, ). To chec, we may calculate the higher order differece sequeces of the cubes maually. Note that we ca calculate the cosecutive cubes (ad, for that matter, the cosecutive th powers, for ay ) usig additio oly, to the exclusio of subtractio ad multiplicatio. Startig with the cubes 0 3 = 0,, 8, 7 ad their differeces we eter the sums + 6 = 8, = 37, = 64 = 4 3 i the 4 th slatig row with slope + (coutig starts with slatig row 0). Similarly, i the ext slatig row we eter the sums = 4, = 6, = 5 = 5 3, etc. We get: 9

12 Thus we ca fid Δ 4 3 = 30 i row, etry 4 (coutig starts with row 0, etry 0). As aother example let us calculate Surj(6, 3). We oly eed the sixth powers 0,, 64, 79: Surj(6, 3) = 79 3(64) + 3 = 73 9 = 540. To chec, we may calculate the sixth powers usig additio oly (just as we did i calculatig the cubes above): From the slatig row prited i red we also see that Surj(6, ) = Δ 0 6 = 6; Surj(6, 4) = Δ = 560; Surj(6, 5) = Δ = 800; Surj(6, 6) = Δ = 70 = 6! We also have the Explicit formula for the umber of surjective fuctios Surj(, ) = ( ) + ( ) ( ) - Clearly, Surj(, 0) = 0. We also have Surj(, ) = for because, i this case, there is oly oe fuctio (the costat fuctio). Recall that the umber of all fuctios from a -set to a -set is. We may survey them accordig to the umber of elemets i the image, to fid that Surj(, ) + Surj(, ) Surj(, ) + 0 Surj(, 0) = This pleasatly trasparet formula ca be checed through direct coutig: Surj(, ) = Surj(, ) + Surj(, ) = Surj(, 3) + 3 Surj(, ) + 3 Surj(, ) = 3 Surj(, 4) + 4 Surj(, 3) + 6 Surj(, ) + 4 Surj(, ) = For example, if = 3, we have (6) + 3(6) + 3() + (0) = 7 = 3 3. As a example, let = 5 ad fid Surj(5, 3). We oly eed the fifth powers 0,, 3, 43. Surj(5,3) = 43 3(3) + 3 = = 50. We also have Surj(5, ) = 3 () = 30. Let us cotiue the calculatio of the fifth powers beyod 43 = 3 5 :

13 From the slatig row prited i red we fid that Surj(5, 4) = Δ = 40, ad Surj(5, 5) = Δ = 0 = 5!. It is importat to ote that the patter exhibited by the Sierpisi s triagles is a cosequece of Newto s formulas ad of the fact that the slatig rows of Pascal s triagle are i arithmetic progressio (see Exercises). I Chapter 6 we shall see the dual of Newto s First ad Secod Formulas uder the ame Vertical ad Horizotal Exchage Formulas. They are dual i the same sese as the lattice of quotiet sets is dual to the lattice of subsets. I passig we draw attetio to the fact that the Theorem above o the solutio of the ifiite system of liear equatios with the biomial coefficiets as coefficiets ca be formulated as follows. The solutio of the matrix equatio x y 0 0. x y 0. x 3 = y x4 y ca be obtaied by the formula x y x 0 0. y x 3 = 0. y 3 x y where i the first ifiite square matrix we have the biomial coefficiets, i the secod, the biomial coefficiets with alteratig sigature. I other words, the product of the ifiite square matrices is the uit square matrix =

14 Exercises. Calculate Surj(7, 5) i three differet ways.. Calculate the seveth powers of itegers up to 0 7 usig additio oly. 3. I how may differet ways ca we distribute eight ulie balls ito five ulie cells i such a way that o cell shall remai empty? 4. I how may ways ca we have a street of te houses paited with two, three, four, five, six, seve, eight, ie, te give colors i such a way that, i each case, every color is represeted? 5. Show, i at least two differet ways, that Surj(, ) =! 6. For which values of does the equality Surj(, ) = 0 hold? 7. Prove = ( ) ( )... ( )! i at least two differet ways. 8. For which values of is the formula ( ) + ( ) +... = 0 valid? 9. Prove the Biomial Theorem i at least three differet ways. 0. Show that the etries i the th slatig row with slope ± of Pascal s triagle are i arithmetic progressio of order. Fid i terms of. Calculate the th differece sequece.. Show that the biomial coefficiets i row p of Pascal s triagle, with the exceptio of the first ad last, are divisible by p provided that p is a prime umber.. Prove the formula ( ) + = + i at least two differet ways. 3. Prove that the umbers a are i arithmetic progressio of order if, ad oly if, the fuctio A() = a is a polyomial of degree i the variable. 4. Costruct the first 40 rows of the Sierpisi triagle (mod 7) for the biomial coefficiets. 5. Prove the Summatio Formula for biomial coefficiets i at least two differet ways. 6. Prove the followig versio of the Biomial Theorem: for p prime, =,, 3,, p ( ) p p a + b a + b (mod p) 3

15 p 7. Prove the Little Fermat Theorem statig that a (mod p), provided that p is a prime umber that does ot divide a, i at least three differet ways. 8. Prove the Geeralized Little Fermat Theorem statig that for a prime umber p that p does t divide a, a (mod p), for =,, 3, 9. Show that the biomial coefficiets i row p of Pascal s triagle, with the exceptio of the first ad the last, are divisible by p provided that p is a prime umber ad =,, 3, Are the previous statemets true of false for composite umbers? If false, provide couter examples. (a) Show that i a arithmetic progressio of order, if a member a is divisible by the prime umber p, the so is a + p. (b) What ca you say about a + p, a + 3p, a + 4p,... uder the same assumptio? (c) Ivestigate the validity of the statemet for a + where m =,, 3, m p. Let a, a, a 3,..., a,... be a arithmetic progressio of order. Suppose that a = p < is a odd prime umber. Show that Δ p a + 0 (mod p).. (a) Show that Surj(p, ) is divisible by p, provided that p is a prime umber ad >. (b) Ivestigate the validity of the above statemet for Surj(p, ), =,, 3,... p 3. Show that (mod p) where p is a prime umber, for =,, 3,... Is this statemet valid for a composite umber? 4. Prove the Theorem o the solutio of the system of liear equatios with ifiitely may uows i aother way: express x, x, x 3, from the st, d, 3 rd, equatio, substitutig these values ito subsequet equatios. 5. Solve umerically the system of liear equatios with ifiitely may uows, ad chec: (a) x = 0 x + x = x + x + x 3 = x + 3x + 3x 3 + x 4 = (b) x = x x = 33

16 x x + x 3 = x 3x + 3x 3 x 4 = (c) x = 0 x + x = x + x + x 3 = 4 x + 3x + 3x 3 + x 4 = where o the right-had side we have the square umbers. (d) x = 0 x + x = x + x + x 3 = 8 x + 3x + 3x 3 + x 4 = where o the right-had side we have the cubes. (e) x = x + x = 0 x + x + x 3 = 00 x + 3x + 3x 3 + x 4 = where o the right-had side we have the powers of 0. (f) x = y x x = y x x + x 3 = y 3 x 3x + 3x 3 x 4 = y where o the left-had side we have the biomial coefficiets with alteratig sigature, ad y, y, y 3, are arbitrary costats. 6. Calculate (mod p) where p is prime, the followig sums: (a) + p 0 p, (b) p + 0 p, (c) p + 0, (d) + p 0. 34

17 7. Solve the matrix equatio x 0 0. x 0. x 3 = x where the etries i the ifiite square matrix are the biomial coefficiets. Write the matrix equatio as a system of liear equatios with ifiitely may uows, ad chec your solutio by substitutio. 8. Complete the colorig of Sierpisi s triagles o p 3, 4. 35