FareyBey

This is a mathematical tool application focused on exploring number theory, covering core concepts such as Farey sequences, Euler functions, and coprime distributions. Users can learn the Farey summation (mediator) operation of fractions through an intuitive visual interface, and observe how the rational number produced by adding the numerator and denominator falls between the original fractions. The application provides an interactive Euler function calculator to verify the mathematical relationship between φ (n) and Farey sequence length | F_n |, i.e. | F_n |=1+∑ φ (i). Users can adjust the sequence order, view the function value distribution chart in real-time, and experience the beauty of number theory. The coprime distribution analysis function demonstrates that the probability of coprime between two random integers approaches 6/π² ≈ 0.6079, and presents the distribution pattern of coprime under different denominators through dynamic charts. In addition, it also includes Diophantine approximation and continuous fraction tools to help understand the optimal approximation method for rational numbers to irrational numbers.