Associated Topics || Dr. Math Home || Search Dr. Math

### Farey Series

```Date: 10/21/2002 at 09:24:35
From: Clive
Subject: Farey's series

Dear Dr. Math,

I have encountered some difficulties in solving problems about Farey's
series.

For 3 successive terms in a Farey's series, say a/b, c/d, e/f, how can
we show that

i) c/d = (a+e)/(b+f)

I can prove that i) implies ii) and ii) implies i), but I cannot prove
either one independently. Any help will be appreciated. Thank you.

Yours sincerely,
Clive
```

```
Date: 12/10/2002 at 01:51:48
From: Doctor Nitrogen
Subject: Re: Farey's series

Hi, Clive:

Here is a suggestion as to how you can prove (i) and (ii)
independently:

[A] Let a <= b <= n,

[B] e <= f <= n for some n.

Since a/b < e/f, a < b, e < f, and since a + e < b + f, it is also
true that

[1]  a/b < (a + e)/(b + f)

(Why? Hint: look at the inequality a(a + e) < b(b + f), and at
(a + e)/(b + f) < b/a). "Manipulate" this inequality around
and see what happens.)

Therefore,

a/b < (a + e)/(b + f).

With a similar argument,

(a + e)/(b + f) < e/f. Therefore

a/b < (a + e)/(b + f) < e/f is true.

Now for proving (ii), let GCD(a + e, b + f) = k => 1. then

(a + e)/(b + f) = kc/kd = c/d, for some integers c and d,
with GCD(c, d) = 1.

There is a theorem in number theory that says if GCD(c, d) = 1, then
the equation

cx + dy = 1 has a solution for (x, y).

Similarly, for GCD(a, b) =1, the equation

bx + dy = 1 has a solution for (x, y),

and the solutions will be

(b, -a), and (c, -d)

respectively (you might have to prove this to convince yourself).

One reference that has a theorem and proof close to this last result
is _The Theory of Numbers: An Introduction_, Anthony A. Gioia, Markham
Publishing. This is an older publication dating back to the sixties.

In {A] and [B] above, the number n has a connection to Farey series.
For example, for the Farey Series

0/1, 1/7, 1/6, 1/5, 1/4, 2/7,

1/3, 2/5, 3/7, 1/2, 4/7, 3/5,

2/3, 5/7, 3/4, 4/5, 5/6, 6/7,

1,

for this Farey series, n = 7.

The conditions [A] and [B] above are actually required for a Farey
series.

- Doctor Nitrogen, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Number Theory
High School Number Theory

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search