Editing Farey sequence (section)

===Farey fractions and the greatest common divisor===
Since the [[Euler's totient function]] is directly connected to the [[Greatest common divisor|gcd]] so is the number of elements in {{mvar|F<sub>n</sub>}},
<math display=block>|F_n| = 1 + \sum_{m=1}^n \varphi(m) = 1+ \sum\limits_{m=1}^{n} \sum\limits_{k=1}^m \gcd(k,m) \cos {2\pi\frac{k}{m}} .</math>

For any 3 Farey fractions {{math|{{sfrac|''a''|''b''}}, {{sfrac|''c''|''d''}}, {{sfrac|''e''|''f''}}}} the following identity between the [[Greatest common divisor|gcd]]'s of the 2x2 [[matrix determinant]]s in absolute value holds:<ref name=TomasGarcia2020>{{cite journal |last=Tomas Garcia |first=Rogelio |url=http://rtomas.web.cern.ch/rtomas/NNTDM-26-3-005-007.pdf|title=Equalities between greatest common divisors involving three coprime pairs| journal=Notes on Number Theory and Discrete Mathematics |date=August 2020 |volume=26 |issue=3|pages=5–7 |doi= 10.7546/nntdm.2020.26.3.5-7 |s2cid=225280271 |doi-access=free  <!-- |access-date=20 January 2022 --> }}</ref>
<math display=block>
  \gcd\left(\begin{Vmatrix} a & c\\b & d \end{Vmatrix}, \begin{Vmatrix} a & e\\b & f \end{Vmatrix} \right)
  = \gcd\left(\begin{Vmatrix} a & c\\b & d \end{Vmatrix}, \begin{Vmatrix} c & e\\d & f \end{Vmatrix} \right)
  = \gcd\left(\begin{Vmatrix} a & e\\b & f \end{Vmatrix}, \begin{Vmatrix} c & e\\d & f \end{Vmatrix} \right)
</math>

<ref name=Tomas2018>{{cite journal |last=Tomas |first=Rogelio |url=https://cs.uwaterloo.ca/journals/JIS/VOL25/Tomas/tomas5.pdf|title=Partial Franel sums | journal=Journal of Integer Sequences | date=January 2022 |volume=25 |issue=1  <!-- |access-date=16 January 2022 --> }}</ref>