MS_Kekas wrote: Represent $\frac{1}{2021}$ as a difference of two irreducible fractions with smaller denominators. From $\frac ab-\frac cd=\frac 1{2021}$, we get $ad-bc=\frac{bd}{2021}$ And so $(b,d)\in\{(2021u,v),(u,2021v),(43u,47v),(47u,43v)\}$ I dont know what is the order relation you use when you speak of "smallest" pair of integers. If we suppose that smallest is the one for which $\max(|b|,|d|)$ is smallest, then we have $(b,d)=(43,47)$ or $(b,d)=(47,43)$ The first gives for example $\boxed{\frac{11}{43}-\frac{12}{47}=\frac 1{2021}}$ and the second $\boxed{\frac{35}{47}-\frac{32}{43}=\frac 1{2021}}$ (and a lot of others). If your order relation is different, just give it to us and we'll adapt the result.