February’s last day and I wanna to write a little bit before to end this short month, so this time let’s take a look at Farey Sequence It’s useful to find rational approximations of irrational numbers, for computer science applications is used as digital image processing and for math contests, there some nice problems about it. So, here we go…

Definition

For any positive integer , the Farey sequence is the sequence of rational numbers with and arranged in increasing order. For instance,

By generalizing:

Where is the Euler’s totient function, defined as the number of integers in range for which the greatest common divisor

Theorem

If , and are three successive terms in a Farey sequence, then

Example

(IMO 1967) Which fraction , where are positive integers less than , is closest to ? Find all digits after the decimal point in the decimal representation of this fraction that coincide with digits in the decimal representation of .

We have that

Because .

The greatest satisfying the equation are . It’s easy to verify using that best approximates among the fractions with . The numbers and coincide up to the fourth decimal digit: Indeed, gives

What about Farey sequences?

By using some basic facts about Farey sequences, one can find that implies because . Of the two fractions and , the later is closer to

Well, No there so much to talk about farey sequence (As part of math contests at least), but if you find problems about it please send me a message and maybe we can try to solve it together. By the way, I found an interesting question in MathExchange, Check out: What is the sum of the squares of the differences of consecutive element of a Farey Sequence?