Concrete mathematics : a foundation for computer science

146 NUMBER THEORY
22 The number 1111111111111111111 is prime. Prove that, in any radix b, Is this a test for
(11 . . . 1 )b can be prime only if the number of 1 ‘s is prime.
strabismus?
23 State a recurrence for p(k), the ruler function in the text’s discussion of
ez(n!). Show that there’s a connection between p(k) and the disk that’s
moved at step k when an n-disk Tower of Hanoi is being transferred in
2" - 1 moves, for 1 < k 6 2n - 1.
24 Express e,(n!) in terms of y,,(n), the sum of the digits in the radix p Look, ma,
representation of n, thereby generaliZing (4.24). sideways addition.
25 We say that m esactly divides n, written m\\n, if m\n and m J- n/m.
For example, in the text’s discussion of factorial factors, p”P(“!)\\n!.
Prove or disprove the following:
a k\\n and m\\n ++ km\\n, if k I m.
b For all m,n > 0, either gcd(m, n)\\m or gcd(m, n)\\n.
26 Consider the sequence I& of all nonnegative reduced fractions m/n such
that mn 6 N For example,
cJIO = 0 11111111 z 1 z i 3 2 5 3 4 s 6 z s 9 lo
1'10'9'8'7'b'5'4'3'5'2'3'1'2'1'2'1'2'1'1'~'1'1'1'1' 1
Is it true that m’n - mn’ = 1 whenever m/n immediately precedes
m//n’ in $Y!N?
27 Give a simple rule for c:omparing rational numbers based on their representations
as L’s and R’s in the Stern-Brocot number system.
28 The Stern-Brocot representation of 7[ is
rr = R3L7R’5LR29i’LRLR2LR3LR14L2R,. . ;
use it to find all the simplest rational approximations to rc whose denominators
are less than 50. Is y one of them?
29 The text describes a correspondence between binary real numbers x =
(.blb2b3.. . )2 in [0, 1) and Stern-Brocot real numbers o( = B1 B2B3 . . . in
[O, 00). If x corresponds to 01 and x # 0, what number corresponds to
l--x?
30 Prove the following statement (the Chinese Remainder Theorem): Let
ml, . . . . m, be integers with mj I mk for 1 6 j < k < r; let m =
ml . . . m,; and let al, . . . . arr A be integers. Then there is exactly one
integer a such that
a=ak(modmk)fOrl