Concrete mathematics : a foundation for computer science

148 NUMBER THEORY
40 If the radix p representation of n is (a,,, . . . al ao)v, prove that
41
42
Wp epCn!) E (-l)“P(n!‘a,!. . . a,! ao! (mod p)
(The left side is simply n! with all p factors removed. When n = p this
reduces to Wilson’s theorem.)
a Show that if p mod. 4 = 3, there is no integer n such that p divides
n* + 1. Hint: Use :Fermat’s theorem.
b But show that if p mod 4 = 1, there is such an integer. Hint: Write
(P - I)! as (II,=, ‘p~‘i’2 k(p - k)) and think about Wilson’s theorem.
Consider two fractions m/n and m//n’ in lowest terms. Prove that when
the sum m/n+m’/n’ is reduced to lowest terms, the denominator will be
nn’ if and only if n I n’. (In other words, (mn’+m’n)/nn’ will already
be in lowest terms if and only if n and n’ have no common factor.)
43 There are 2k nodes at level k of the Stern-Brocot tree, corresponding to
the matrices Lk Lkp’ R ..I Rk. Show that this sequence can be obtained
by starting with Lk and’then multiplying successively by
0 -1
1 2p(n) + 1 >
for 1 6 n < 2k, where p(n) is the ruler function.
44 Prove that a baseball player whose batting average is .316 must have
batted at least 19 times. (If he has m hits in n times at bat, then
m/n E [.3155, .3165).)
45 The number 9376 has the peculiar self-reproducing property that
9376* = 87909376
How many 4-digit numbers x satisfy the equation x2 mod 10000 = x?
How many n-digit numbers x satisfy the equation x2 mod 10n = x?
46 a Prove that if nj = l and nk = 1 (mod m), then nscd(jtk) = 1.
b Show that 2” f 1 (mod n), if n > 1. Hint: Consider the least prime
factor of n.
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48
Show that if nmp’ E 1 (mod m) and if n(“-‘)/p $ 1 (mod m) for all
primes such that p\(m - l), then m is prime. Hint: Show that if this
condition holds, the numbers nk mod m are distinct, for 1 6 k < m.
Generalize Wilson’s theorem (4.49) by ascertaining the value of the expression
u-I1