Concrete mathematics : a foundation for computer science

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504 ANSWERS TO EXERCISES 4.40 Let f(n) = n,,,,,,,,, k = n!/pl"/pJ Ln/p]! and g(n) = n!/pEP(“!l. Then s(n) = f(n)f( ln/PJ) f( ln/p’J) . . . = f(n) g( b/d) . Also f(n) = ao!(p - l)!Ln/pl = ao!(-l)L"/PJ (mod p), and e,(n!) = Ln/pJ + cp (Ln/pJ !) . These recurrences make it easy to prove the result by induction. (Several other solutions are possible.) 4.41 (a) If n2 = -1 (mod p) then (n2)(pP’i/2 = -1; but Fermat says it’s +l. (b) Let n = ((p - 1)/2)!; we have n = (--l)(P~‘i’2 n,sk n.1 or rn2 < n2 do if rnz < n2 then (output(L); (nl,nz) := (nl,nz) - (ml,m,)) else (output(R); (ml, m2) := (ml, ml) - (nl ,nz)) . The output is LLLRRRRR = & z .3158. Incidentally, an average of .334 implies at least 287 at bats.
A ANSWERS TO EXERCISES 505 4.45 x2 E x (mod 10n) ( x(x - 1) E 0 (mod 2”) and x(x - 1) E 0 (mod 5n) M x mod 2” = [Oor 11 and x mod 5” = [Oor 11. (The last step is justified because x(x - 1) mod 5 = 0 implies that either x or x - 1 is a multiple of 5, in which case the other factor is relatively prime to 5n and can be divided from the congruence.) So there are ,at most four solutions, of which two (x = 0 and x = 1) don’t qualify for the title “n-digit number” unless n = 1. The other two solutions have the forms x and 1 On + 1 -- x, and at least one of these numbers is > 1 On-‘. When n = 4 the other solution, 10001 - 9376 = 625, is not a four-digit number. 1Ne expect to get two n-digit solutions for about 90% of all n, but this conjecture has not been proved. (Such self-reproducing numbers have been called “automorphic.“) 4.46 (a) If j’j - k’k = gcd(j,k), we have nk’knscdii,k) = ni’i = 1 and nk’k - 1. (b) L e tn = pq, where p is the smallest prime divisor of n. If 2” E 1 (mod n) then 2” G 1 (mod p). Also 2P-l = 1 (mod p); hence 2scdipm ‘,nl = 1 (mod p). But gcd(p - 1 In) = 1 by the definition of p. 4.47 If n+’ = 1 (mod m) we must have n I m. If nk = nj for some 1 < j < k < m, then nkPj = 1 because we can divide by nj. Therefore if the numbers n’ mod m, . , n”-’ mod m are not distinct, there is a k < m - 1 with nk = 1. The least such k divides m- 1, by exercise 46(a). But then kq = (m - 1 )/p for some prime p and some positive integer q; this is impossible, since nkq $ 1. Therefore the numbers n’ mod m, . . , nmP’ mod m are distinct and relatively prime to m. Therefore the numbers 1, . , m - 1 are relatively prime to n-L, and m must be prime. 4.48 By pairing numbers up with their inverses, we can reduce the product (mod m) to n l~n 2); otherwise it’s +l. 4.49 (a) Either m < n (@(N - 1) cases) or m = n (one case) or m > n (O(N - 1) again). Hence R(N) = 2@(N - 1) + 1. (b) From (4.62) we get 2@(N-l)+l = l+x~(d)LN/d]LN/d-11; d>l hence the stated result holds if and only if x p(d) LN/dJ = 1 , d2’ for N :z 1 And this is a special case of (4.61) if we set f(x) = (x 3 1)
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CONCRETE MATHEMATICS Dedicated to L
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CONCRETE MATHEMATICS Ronald L. Grah
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“A odience, level, and treatment
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‘I a concrete life preserver thro
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I’m unaccustomed to this face. De
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n [ n-l 1 n {I m Prestressed concre
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CONTENTS xiii 5.3 Tricks of the Tra
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2 RECURRENT PROBLEMS It’s not imm
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4 RECURRENT PROBLEMS Of course the
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6 RECURRENT PROBLEMS through any of
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8 RECURRENT PROBLEMS From these sma
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10 RECURRENT PROBLEMS But what abou
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12 RECURRENT PROBLEMS that is, in t
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14 RECURRENT PROBLEMS by separating
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16 RECURRENT PROBLEMS For example,
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18 RECURRENT PROBLEMS Homework exer
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20 RECURRENT PROBLEMS 21 Suppose th
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22 SUMS The three-dots notation has
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24 SUMS A slightly modified form of
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26 SUMS where A(n), B(n), and C(n)
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28 SUMS Let’s apply these ideas t
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30 SUMS 2.3 MANIPULATION OF SUMS No
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32 SUMS Typically we use rule (2.20
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34 SUMS because the derivative of a
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36 SUMS instead of summing over all
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38 SUMS The second sum here is zero
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40 SUMS The normal way to evaluate
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42 SUMS First, as usual, we look at
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44 SUMS order to get an equation fo
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46 SUMS One way to use this fact is
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48 SUMS definition where the factor
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50 SUMS Let’s try to recap this;
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52 SUMS we need an appropriate defi
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54 SUMS This formula indicates why
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56 SUMS u(x) = x and Av(x) = 2’;
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58 SUMS The definition in the previ
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60 SUMS In fact, our definition of
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62 SUMS tCj,kiEF a. J, k < A for al
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64 SUMS 18 Let 9%~ and Jz be the re
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66 SUMS 34 35 36 Prove that if the
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68 INTEGER FUNCTIONS below the line
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70 INTEGER FUNCTIONS same inequalit
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72 INTEGER FUNCTIONS An important s
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74 INTEGER FUNCTIONS By the way, th
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76 INTEGER FUNCTIONS (Previously K
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78 INTEGER FUNCTIONS This derivatio
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80 INTEGER FUNCTIONS We’ve got mo
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82 INTEGER FUNCTIONS division, and
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84 INTEGER FUNCTIONS more than one
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86 INTEGER FUNCTIONS 3.5 FLOOR/CEIL
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88 INTEGER FUNCTIONS For this purpo
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90 INTEGER FUNCTIONS it comes out i
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92 INTEGER FUNCTIONS To summarize,
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94 INTEGER FUNCTIONS Also, as we gu
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96 INTEGER FUNCTIONS Basics 10 Show
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98 INTEGER FUNCTIONS 31 Prove or di
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100 INTEGER FUNCTIONS 43 Find an in
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4 Number Theory INTEGERS ARE CENTRA
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104 NUMBER THEORY The left side can
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106 NUMBER THEORY product of primes
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108 NUMBER THEORY only finitely man
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110 NUMBER THEORY But 2P - 1 isn’
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112 NUMBER THEORY arrangements in a
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114 NUMBER THEORY The binary repres
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116 NUMBER THEORY A fraction m/n is
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118 NUMBER THEORY A mediant fractio
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120 NUMBER THEORY This representati
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122 NUMBER THEORY This means that w
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124 NUMBER THEORY Since x mod m dif
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126 NUMBER THEORY than to say that
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128 NUMBER THEORY require division,
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130 NUMBER THEORY Now we must show
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132 NUMBER THEORY And it’s possib
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134 NUMBER THEORY For example, (~(1
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136 NUMBER THEORY Conversely, if f(
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138 NUMBER THEORY (We must add 1 to
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140 NUMBER THEORY The problem of co
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142 NUMBER THEORY n is odd, n4 and
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144 NUMBER THEORY But the inner sum
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146 NUMBER THEORY 22 The number 111
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148 NUMBER THEORY 40 If the radix p
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150 NUMBER THEORY 57 Let S(m,n) be
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152 NUMBER THEORY 73 If the 0(n) +
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154 BINOMIAL COEFFICIENTS For examp
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156 BINOMIAL COEFFICIENTS And now t
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158 BINOMIAL COEFFICIENTS property
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160 BINOMIAL COEFFICIENTS (3, then
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162 BINOMIAL COEFFICIENTS Binomial
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164 BINOMIAL COEFFICIENTS Table 164
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166 BINOMIAL COEFFICIENTS not rows.
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168 BINOMIAL COEFFICIENTS Equation
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170 BINOMIAL COEFFICIENTS that m =
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172 BINOMIAL COEFFICIENTS ifweuse (
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176 BINOMIAL COEFFICIENTS sum on th
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178 BINOMIAL COEFFICIENTS some data
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180 BINOMIAL COEFFICIENTS Problem 4
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182 BINOMIAL COEFFICIENTS We’re l
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184 BINOMIAL COEFFICIENTS We can’
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186 BINOMIAL COEFFICIENTS 5.3 TRICK
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188 BINOMIAL COEFFICIENTS if we app
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190 BINOMIAL COEFFICIENTS The Newto
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192 BINOMIAL COEFFICIENTS One examp
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194 BINOMIAL COEFFICIENTS We can de
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196 BINOMIAL COEFFICIENTS This obse
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198 BINOMIAL COEFFICIENTS Generatin
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200 BINOMIAL COEFFICIENTS can be pu
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204 BINOMIAL COEFFICIENTS This hold
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206 BINOMIAL COEFFICIENTS It’s ou
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208 BINOMIAL COEFFICIENTS into this
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210 BINOMIAL COEFFICIENTS integer.
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212 BINOMIAL COEFFICIENTS into to b
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214 BINOMIAL COEFFICIENTS Now sin(
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216 BINOMIAL COEFFICIENTS But look
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218 BINOMIAL COEFFICIENTS It’s al
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220 BINOMIAL COEFFICIENTS A similar
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222 BINOMIAL COEFFICIENTS One use o
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224 BINOMIAL COEFFICIENTS Therefore
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226 BINOMIAL COEFFICIENTS Now f(-b)
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228 BINOMIAL COEFFICIENTS negative
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230 BINOMIAL COEFFICIENTS valid for
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232 BINOMIAL COEFFICIENTS 18 Find a
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234 BINOMIAL COEFFICIENTS 37 Show t
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236 BINOMIAL COEFFICIENTS 57 Use Go
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238 BINOMIAL COEFFICIENTS 72 Prove
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240 BINOMIAL COEFFICIENTS 86 Let al
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242 BINOMIAL COEFFICIENTS Research
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244 SPECIAL NUMBERS Table 244 Stirl
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246 SPECIAL NUMBERS There are eleve
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248 SPECIAL NUMBERS cycle arrangeme
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250 SPECIAL NUMBERS Table 250 Basic
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252 SPECIAL NUMBERS We can remember
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254 SPECIAL NUMBERS Table 254 Euler
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{'l 'I L\‘, 0 1 / 2 256 SPECIAL N
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258 SPECIAL NUMBERS Table 258 Stirl
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260 SPECIAL NUMBERS ’ ’ ’ (Th
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262 SPECIAL NUMBERS we found that a
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264 SPECIAL NUMBERS Rearranging, we
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266 SPECIAL NUMBERS Thus we have th
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268 SPECIAL NUMBERS (We have used t
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270 SPECIAL NUMBERS Bernoulli numbe
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272 SPECIAL NUMBERS trigonometric f
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274 SPECIAL NUMBERS Recurrence (6.9
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276 SPECIAL NUMBERS Gnethingwecando
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278 SPECIAL NUMBERS cases are: a0 =
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280 SPECIAL NUMBERS The process of
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282 SPECIAL NUMBERS Fk < n < Fk+l;
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284 SPECIAL NUMBERS We have now boi
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286 SPECIAL NUMBERS Before we stop
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288 SPECIAL NUMBERS The continuant
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290 SPECIAL NUMBERS in four steps:
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292 SPECIAL NUMBERS Thus we can con
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294 SPECIAL NUMBERS numbers are 1,
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296 SPECIAL NUMBERS 6 An explorer h
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298 SPECIAL NUMBERS 24 Prove that t
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300 SPECIAL NUMBERS 44 Prove the co
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302 SPECIAL NUMBERS 61 Prove the id
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304 SPECIAL NUMBERS 80 Show that co
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7 Generating Functions THE MOST POW
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308 GENERATING FUNCTIONS are added
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310 GENERATING FUNCTIONS (The last
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312 GENERATING FUNCTIONS Now we hav
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314 GENERATING FUNCTIONS The first
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316 GENERATING FUNCTIONS determine
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318 GENERATING FUNCTIONS operation
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320 GENERATING FUNCTIONS Table 320
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322 GENERATING FUNCTIONS middle of
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324 GENERATING FUNCTIONS Fortunatel
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326 GENERATING FUNCTIONS Once we’
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328 GENERATING FUNCTIONS Step 1 is
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330 GENERATING FUNCTIONS nice prope
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332 GENERATING FUNCTIONS Finally, s
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334 GENERATING FUNCTIONS Step 3 was
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336 GENERATING FUNCTIONS This is a
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338 GENERATING FUNCTIONS Table 337
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340 GENERATING FUNCTIONS F(z)' = i
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342 GENERATING FUNCTIONS We can app
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344 GENERATING FUNCTIONS Step 3 is
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346 GENERATING FUNCTIONS Raney’s
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348 GENERATING FUNCTIONS Notice tha
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350 GENERATING FUNCTIONS Chapter 5
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352 GENERATING FUNCTIONS Now let’
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354 GENERATING FUNCTIONS function f
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356 GENERATING FUNCTIONS 7.7 DIRICH
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358 GENERATING FUNCTIONS 5 Find a g
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360 GENERATING FUNCTIONS 20 A power
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362 GENERATING FUNCTIONS 29 What is
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364 GENERATING FUNCTIONS 43 The New
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366 GENERATING FUNCTIONS 53 The seq
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368 DISCRETE PROBABILITY total of 6
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370 DISCRETE PROBABILITY about R if
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372 DISCRETE PROBABILITY The mean o
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374 DISCRETE PROBABILITY more sprea
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376 DISCRETE PROBABILITY hence VS =
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378 DISCRETE PROBABILITY And we can
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380 DISCRETE PROBABILITY variance b
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382 DISCRETE PROBABILITY (1 +z+...
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384 DISCRETE PROBABILITY ~1 and ~2
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386 DISCRETE PROBABILITY where Ug i
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388 DISCRETE PROBABILITY Repeating
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390 DISCRETE PROBABILITY There’s
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392 DISCRETE PROBABILITY But the co
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394 DISCRETE PROBABILITY In the spe
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396 DISCRETE PROBABILITY Here 1 is
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398 DISCRETE PROBABILITY For exampl
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400 DISCRETE PROBABILITY Thus the m
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402 DISCRETE PROBABILITY Let sj be
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404 DISCRETE PROBABILITY Therefore
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406 DISCRETE PROBABILITY The mean v
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408 DISCRETE PROBABILITY Complicate
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410 DISCRETE PROBABILITY all k and
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412 DISCRETE PROBABILITY to be a pg
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414 DISCRETE PROBABILITY 10 What’
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416 DISCRETE PROBABILITY 28 What is
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418 DISCRETE PROBABILITY 38 What is
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420 DISCRETE PROBABIL1T.Y a Once he
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422 DISCRETE PROBABILITY 50 Conside
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424 DISCRETE PROBABILITY 58 Are the
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426 ASYMPTOTICS The word asymptotic
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428 ASYMPTOTIC3 How about the funct
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430 ASYMPTOTICS N. G. de Bruijn beg
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432 ASYMPTOTICS subtle consideratio
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434 ASYMPTOTICS inefficient “beca
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436 ASYMPTOTICS 9.3 0 MANIPULATION
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138 ASYMPTOTICS Table 438 Asymptoti
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440 ASYMPTOTICS Notice how the 0 te
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442 ASYMPTOTICS Problem 3: The nth
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444 ASYMPTOTICS Problem 4: A sum fr
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446 ASYMPTOTICS Problem 5: An infin
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448 ASYMPTOTICS We might as well do
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450 ASYMPTOTIC3 a rough estimate an
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452 ASYMPTOTICS This last estimate
Page 468 and 469: 454 ASYMPTOTICS Here’s another ex
Page 470 and 471: 456 ASYMPTOTICS On the left is a ty
Page 472 and 473: 458 ASYMPTOTICS (The Bernoulli poly
Page 474 and 475: 460 ASYMPTOTICS The graph of B,(x)
Page 476 and 477: 462 ASYMPTOTICS 9.6 FINAL SUM:MATIO
Page 478 and 479: 464 ASYMPTOTICS The integral JT B4(
Page 480 and 481: 466 ASYMPTOTIC3 where 0 < a,,,, < 1
Page 482 and 483: 468 ASYMPTOTICS In Chapter 5, equat
Page 484 and 485: 470 ASYMPTOTICS For example, we hav
Page 486 and 487: 472 ASYMPTOTICS The final task seem
Page 488 and 489: 474 ASYMPTOTICS This is our approxi
Page 490 and 491: 476 ASYMPTOTIC23 9 Prove (9.22) rig
Page 492 and 493: 478 ASYMPTOTICS 39 Evaluate xOsk
Page 494 and 495: 480 ASYMPTOTICS a What is the asymp
Page 496 and 497: 482 ASYMPTOTIC3 Research problems 6
Page 498 and 499: 484 ANSWERS TO EXERCISES Venn [294]
Page 500 and 501: 486 ANSWERS TO EXERCISES move the b
Page 502 and 503: 488 ANSWERS TO EXERCISES 2.2 This i
Page 504 and 505: 490 ANSWERS TO EXERCISES 2.24 Summi
Page 506 and 507: 492 ANSWERS TO EXERCISES 2.36 (a) B
Page 508 and 509: 494 ANSWERS TO EXERCISES 3.21 If 10
Page 510 and 511: 496 ANSWERS TO EXERCISES the circle
Page 512 and 513: 498 ANSWERS TO EXERCISES 3.40 Let L
Page 514 and 515: 500 ANSWERS TO EXERCISES 3.50 H. S.
Page 516 and 517: 502 ANSWERS TO EXERCISES 4.22 (b,
Page 520 and 521: 506 ANSWERS TO EXERCISES 4.50 (a) I
Page 522 and 523: 508 ANSWERS TO EXERCISES and (k- 1)
Page 524 and 525: 510 ANSWERS TO EXERCISES In both ca
Page 526 and 527: 512 ANSWERS TO EXERCISES 5.7 Yes, b
Page 528 and 529: 514 ANSWERS TO EXERCISES because al
Page 530 and 531: 516 ANSWERS TO EXERCISES 5.44 Cance
Page 532 and 533: 518 ANSWERS TO EXERCISES and this i
Page 534 and 535: 520 ANSWERS TO EXERCISES The infini
Page 536 and 537: 522 ANSWERS TO EXERCISES 5.82 Let ~
Page 538 and 539: 524 ANSWERS TO EXERCISES Since ~06j
Page 540 and 541: 526 ANSWERS TO EXERCISJES 5.94 This
Page 542 and 543: 528 ANSWERS TO EXERCISES 6.21 The h
Page 544 and 545: 530 ANSWERS TO EXERCISIES 6.40 If 6
Page 546 and 547: 532 ANSWERS TO EXERCISES solutions
Page 548 and 549: 534 ANSWERS TO EXERCISES 6.62 (a) A
Page 550 and 551: 536 ANSWERS TO EXERCISES 6.74 If p(
Page 552 and 553: 538 ANSWERS TO EXERCISIES One of th
Page 554 and 555: 540 ANSWERS TO EXERCISES 6.85 The p
Page 556 and 557: 542 ANSWERS TO EXERCISES Hence gzn+
Page 558 and 559: 544 ANSWERS TO EXERCISES 7.24 ntk,+
Page 560 and 561: 546 ANSWERS TO EXERCISES 7.40 The e
Page 562 and 563: 548 ANSWERS TO EXERCISES 1 + a. Hen
Page 564 and 565: 550 ANSWERS TO EXERCISES applied to
Page 566 and 567: 552 ANSWERS TO EXERCISES 8.13 (Solu
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554 ANSWERS TO EXERCISES 8.24 (a) A
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556 ANSWERS TO EXERCISES 8.32 By sy
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558 ANSWERS TO EXERCISES 8.37 The n
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560 ANSWERS TO EXERCISES probabilit
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562 ANSWERS TO EXERCISES And since
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564 ANSWERS TO EXERCISES 8.56 If m
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566 ANSWERS TO EXERCISES We can now
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568 ANSWERS TO EXERCISES 9.19 Hlo =
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570 ANSWERS TO EXERCIS:ES 9.30 Let
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572 ANSWERS TO EXERCISES 9.42 The h
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574 ANSWERS TO EXERCISIES length of
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576 ANSWERS TO EXERCISES 9.58 Let 0
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B Bibliography HERE ARE THE WORKS c
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580 BIBLIOGRAPHY 24' 25 26 27 28 29
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582 BIBLIOGRAPHY 51 52 53 54 55 56
Page 598 and 599:
584 BIBLIOGRAPHY 78 79 80 81 82 83
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586 BIBLIOGRAPHY 100 R. A. Fisher,
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588 BIBLIOGRAPHY 128 Ronald L. Grah
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590 BIBLIOGRAPHY 160 K. Inkeri, “
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592 BIBLIOGRAPHY 188 E.E. Kummer,
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594 BIBLIOGRAPHY 214 Z.A. Melzak, C
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596 BIBLIOGRAPHY 243 George N. Rane
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598 BIBLIOGRAPHY 271 A. D. Solov’
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600 BIBLIOGRAPHY 299 Louis Weisner,
Page 616 and 617:
602 CREDITS FOR EXERCISES 1.1 1.2 1
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604 CREDITS FOR EXERCISES 6.46 6.47
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Index WHEN AN INDEX ENTRY refers to
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608 INDEX Bois-Reymond, Paul David
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610 INDEX Degenerate hypergeometric
Page 626 and 627:
612 INDEX Feder, Tom&s, 604. Feigen
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614 INDEX Hall, Marshall, Jr., 588.
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616 INDEX Lincoln, Abraham, 387. Li
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618 INDEX Patashnik, Oren, iii, iv,
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620 INDEX Running time, 411-412. Ru
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622 INDEX Sylvester, James Joseph,
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List of Tables Sums and differences
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Concrete mathematics : a foundation for computer science

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