On a Theorem of Legendre on Diophantine Approximation

Jaroslav Hančl, Tho Phuoc Nguyen

Legendre's theorem states that every irreducible fraction $\frac{p}{q}$ which satisfies the inequality $\left|\alpha -\frac{p}{q}\right|<\frac{1}{2q^2}$ is convergent to $\alpha$. Later…