Concrete mathematics : a foundation for computer science

280 SPECIAL NUMBERS
The process of reducing Fn*k to a combination of F, and F,+, by using
(6.105) and (6.104) leads to the sequence of formulas
F n+2 = F,+I + F, Fn-I = F,+I - F,
F n+3 = 2F,+, + F, Fn-2 = -F,+, +2F,
F n+4 = 3F,+1 + 2F, Fn-3 = 2F,+, -SF,
F n+5 = 5F,+1 + 3F, Fn-4 = -3F,+, + 5F,
in which another pattern becomes obvious:
F n+k = FkFn+l + h-IF,, . (6.108)
This identity, easily proved by induction, holds for all integers k and n (positive,
negative, or zero).
If we set k = n in (6.108), we find that
F2n = FnFn+l + Fn-I Fn ; (6-g)
hence Fz,, is a multiple of F,. Similarly,
F3n = FznFn+tl + F2n-1Fn,
and we may conclude that F:+,, is also a multiple of F,. By induction,
Fkn is a multiple of F, , (6.110)
for all integers k and n. This explains, for example, why F15 (which equals
610) is a multiple of both F3 and F5 (which are equal to 2 and 5). Even more
is true, in fact; exercise 27 proves that
.wWm, Fn) = Fgcd(m,n) .
(6.111)
For example, gcd(F,Z,F,s) = gcd(144,2584) = 8 = Fg.
We can now prove a converse of (6.110): If n > 2 and if F, is a multiple of
F,, then m is a multiple of n. For if F,\F, then F,\ gcd(F,, F,) = Fgcd(m,n) <
. .
F,. This 1s possible only if Fgcd(m,nl = F,; and our assumption that n > 2
makes it mandatory that gcd(m, n) = n. Hence n\m.
An extension of these divisibility ideas was used by Yuri Matijasevich in
his famous proof [213] that there is no algorithm to decide if a given multivariate
polynomial equation with integer coefficients has a solution in integers.
Matijasevich’s lemma states that, if n > 2, the Fibonacci number F, is a
multiple of F$ if and only if m is a multiple of nF,.
Let’s prove this by looking at the sequence (Fk, mod F$) for k = 1, 2,
3 I “‘, and seeing when Fk,, mod Fi = 0. (We know that m must have the