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36 1. Fundamentals

We can answer Fibonacci’s question by computing each F,, in turn:

1, 1,2,3,5,8,13,21,34,55,89,144,233,377,610,....

The number of pairs present at the beginning of the 13th month is Fl3 =

233, so that after one year, the original pair is responsible for the production

of 232 new ones.

We will look at the question of finding a general formulae for the terms

of the sequence in Exploration E.50. Our interest here is to construct a

polynomial of two variables whose positive values are precisely the numbers

F,, where n is a positive integer.

(a) Show that for each value of n exceeding 1

F,,+mlF,,-l -F,” =

1

1 if 72 is even

-1 if n is odd.

(b) Let x and y be positive integers such that

I(y - x)y - x21 = 1. (*)

It can be shown that y - z, t, y are consecutive terms in the Fibonacci

sequence. First note that x 5 y and y - x 5 x, and that, if x = y, then

(x, y) = (1,1) and, if y - z = 2, then (x,y) = (1,2).

The desired result can be proved by induction. It holds for y 5 F3.

Assume that n > 3 and it holds for y 5 F,. Now let (*) be valid when

x>OandF,,<y<F,+1. Show that x 5 F,, and that, if z = y - x, then

1(x - %)X - %21 = 1.

Use the induction hypothesis to argue that z - z, z, x, and hence z, x, y

are consecutive Fibonacci numbers.

(c) What are the positive values assumed by the polynomial

2 -[(y- x)y- x212

when x and y are integers?
(d) Determine a polynomial f(x, y) with integer coefficients such that,
whenever x and y are integers for which f(x, y) > 0, f(z, y) belongs to the
Fibonacci sequence.

1.7 Rings and Fields

Problems involving polynomials often require us to distinguish whether
the coefficients are rational or nonrational, real or complex. The solution
of even real equations require us to draw in nonreal entities. Since there are
rules of operation equally valid for the various number systems-rational,