Some special property of Farey sequence

We discuss some special property of the Farey sequence. We show in each term of the Farey sequence, ratio of the sum of elements in the denominator and the sum of elements in the numerator is exactly two. We also show that the Farey sequence contains a palindromic structure.


Introduction
Farey sequence is named after British geologist John Farey, Sr., who publised a result in Philosophical Magazine in 1816 about these sequence without giving a proof. Later, Cauchy proved the result conjectured by Farey. Though, Charles Haros proved similar result in 1802 which were not known to Farey and Cauchy. Later, Farey sequence appeared in many different areas of mathematics, including number theory, topology, geometry, see, [5], [3], [1]. Farey sequence is also related to Fords circle and Riemann hypothesis, [6].
Farey sequence [5], F n in the interval [0, 1] is defined as sequence of ascending rational numbers in reduced form starting at 0 1 and ending at 1 1 such that the elements in the nth term, denominator are less than or equal to n. First few terms of Farey sequence are listed as follows, Suppose a b and c d are two fractions in Farey sequence. The mediant or sum of a b and c d is defined as [1]. For all the terms of the Farey sequence given above, one can note that sum of the elements in the denominator is always two times of the sum of the elements in the numerator. In F 1 , sum of numerator is 1 and sum of denominator is 2. In F 2 , sum of numerator is 2 and sum of denominator is 4. Likewise in F 3 , F 4 , F 5 , sum of denominator is always two times the sum of numerator. So, one may ask if this result holds for all F n or not. In this paper, we will show that in general this result is true. Surprisingly, no one has noticed this special property of Farey sequence. In the second section of this article, we introduce this special property of Farey sequence. In the final section, we show that denominators of each fraction in F n has a palindromic structure for all n.

A special property of Farey sequence
In this section, we prove our first result on Farey sequence.

Lemma. [1] The length of each term of Farey sequence F n is given by the following recurring formula:
Where φ is the Euler's Totient function, which is number of elements less than and co-prime to n.

2.2.
Remark. From the Lemma 2.1, we get the information that number of new elements appears in F n compare to F n−1 is φ(n). The numbers 1 ≤ k < n which are co-prime to n, appears to the numerator in the new terms of F n and the denominator of each new terms in F n is always n.

Lemma. [2]
For any positive intergers n and k, Proof. If k is co-prime to n, then n − k is also co-prime to n. Note that, k can not be equal to n − k, otherwise, gcd(k, n) will not be 1. The number of elements co-prime to n is φ(n). So pairing k and n − k, we get the total sum is nφ(n) 2 .

Theorem.
In each term of the Farey sequence, the sum of the elements in the denominator is always two times of the sum of elements in the numerator.
Proof. We prove this result by induction on n. Let N n denotes the sum of the elements in the numerator of the nth term of Farey sequence, that is, in F n and D n denotes the sum of elements in the denominator of nth term of Farey sequence. For n = 1, So, our claim is true for n = 1. Now suppose result is true for n − 1. We show that result is true for all n, we have, k, using remark 2.2.
So, by induction our result follows.

Palindromic structure in Farey sequence
In this final section, we show that there is a palindromic structure in denominators of F n for all n. The following lemma gives a condition for two fraction to be Farey neighbour.

neighbours in the Farey sequence if and only if bc
< · · · ≤ 1 1 are also Farey neighbours.
Proof. Using lemma 3.3, on the given first Farey neighbours, we have, Thus, our corollary follows.

Theorem. Denominators of each fraction in F n for all n in a Farey sequence is a palindrom.
Proof. We prove this theorem by induction.
In F 1 , denominators are 1,1, which is a palindromic sequence. In F 2 , denominators are 1,2,1, which is also a palindromic sequence. Now, suppose that denominators in F n−1 are in a palindromic sequence. We need to show that denominators in F n is also in a palindromic sequence. Using corollary 3.4 and palindromic structure of denominators in F n−1 , we can write F n−1 in the following form, Suppose in the next term F n , a new term appears between a 1 b 1 and a 2 b 2 . So, we can write F n from F n−1 as, Let a 1 + a 2 = k. We have following relations by using remark 2.2, b 1 + b 2 = n and gcd(n, k) = 1.
Thus, F n can be written in terms of n, k as follows: We know that if gcd(n, k) = 1 then gcd(n, n − k) = 1. Now, we need to show are Farey neighbours. So, we need to only check the condition of lemma 3.3. As a 1 b 1 < k n < a 2 b 2 are Farey neighbours, we have, kb 1 − na 1 = 1 and na 2 − kb 2 = 1. (3.2) Note that, are Farey neighbours. Thus, we have proved that denominators in F n are palindrom. Our desired result follows from the induction.
3.6. Remark. By length of a palindrom, we mean the number of element apearing in the palindrom. From the lemma 2.1, it is clear that palindromic length of denominator palindrom in F n is |F n−1 | + φ(n). Thus, using Farey sequence one can get very large palindrom numbers.