Concrete mathematics : a foundation for computer science

150 NUMBER THEORY
57 Let S(m,n) be the set of all integers k such that
mmodk+nmodk 3 k.
For example, S(7,9) = {2,4,5,8,10,11,12,13,14,15,16}. Prove that
x q(k) = m.n
kESlm,n)
Hint: Prove first that x,6msn ,&,,, v(d) = IL>, v(d) ln/dJ. Then
consider L(m + n)/d] - [m/d] - Ln/dJ.
58 Let f(m) = Ed,,,, d. Fi:nd a necessary and sufficient condition that f(m)
is a power of 2.
Bonus problems
59 Prove that if x1, . . . , x, are positive integers with 1 /x1 f. . . + 1 /x, = 1,
then max(xl,. . . ,x,) < e,. Hint: Prove the following stronger result by
induction: “If 1 /x1 +. . . + 1 /x, + l/o1 = 1, where x1, . . . , x, are positive
integers and 01 is a rational number 3 max(xl , . . , xn), then a+ 1 < e,+l
and x1 . xn (a + 1) < el . . . e,e,+l .” (The proof is nontrivial.)
60 Prove that there’s a constant P such that (4.18) gives only primes. You
may use the following (Ihighly nontrivial) fact: There is a prime between
p and p + cp’, for some constant c and all sufficiently large p, where
g=losl.
1920
61 Prove that if m/n, m’/n’, and m/‘/n” are consecutive elements of 3~,
then
m” = [(n+N)/n’]m’-m,
n” = [(n+N)/n’jn’-n.
(This recurrence allows us to compute the elements of 3N in order, starting
with f and ft.)
62 What binary number corresponds to e, in the binary tf Stern-Brocot
correspondence? (Express your answer as an infinite sum; you need not
evaluate it in closed form.)
63 Show that if Fermat’s Last Theorem (4.46) is false, the least n for which
it fails is prime. (You may assume that the result holds when n = 4.)
Furthermore, if aP + bP = cp and a I b, show that there exists an integer
m such that
a+b = mp, if p$c;
pPV1 mP , if p\c.
Thus c must be really huge. Hint: Let x = a + b, and note that
gcd(x, (ap + (x - a)p)/x) = gcd(x,paP-‘).