Concrete mathematics : a foundation for computer science

I have discovered a
wonderful proof of
Fermat’s Last Theorem,
but there’s no
room for it here.
Therefore, if Fermat’s
Last Theorem
is false, the universe
will not be big
enough to write
down any numbers
that disprove it.
A ANSWERS TO EXERCISES 509
Incidentally, this exercise implies that (m + m”)/(n + n”) = m//n’,
although the former fraction is not always reduced.
4.62 2 ‘$2 2+2 3-2 6-2 7+2~'2+2~~'3-2~20-2~21+2~30+2~31-
2 ~42 - 2 43 + . . . can be written
; + 3 t(2-4k1-6k-3 _ 2-4k2-10k -7)
k>O
Incidentally, this sum can be expressed in closed form using the “theta function”
O(z, h) = tk e~xhkz+2irk; we have
e t-3 i + ~;6(~ln2,3iln2) - &O(%ln2,5iln2)
4.63 Any n > 2 either has a prime divisor d or is divisible by d = 4. In either
case, a solution with exponent n implies a solution (an/*)*+(bn/*)* = (c”/*)*
with exponent d. Since d = 4 has no solutions, d must be prime.
The hint follows from the binomial theorem, since aP+(x-a)P-pap ’
is a multiple of x when p is odd. Assume that a -L x. If x is not divisible
by p, x is relatively prime to cP/x; hence x = mp for some m. If x is divisible
by p, then cp/x is divisible by p but not by p2, and cp has no other factors
in common with x.
(The values of a, b, c must, in fact, be even higher than this result
indicates! Inkeri [160] has proved that
A sketch of his proof appears in [249, pages 228-2291, a book that contains
an extensive survey of progress on Fermat’s Last Theorem.)
4.64 Equal fractions in YN appear in “organ-pipe order”:
2m 4m rm 3m m
- - - -
2n’ 4n’ . . . . ml . . . . 3n’ n.
Suppose that IPN is correct; we want to prove that &+I is correct. This
means that if kN is odd, we want to show that
k-l
N+l
= yN,kN;
if kN is even, we want to show that
k-l
yN,kN 1 yN,kN ~ N+l
yN,kN YN,kN+l *