Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Apéry-like numbers. In this paper, we use Beukers’ method to prove some conjectures of the first author concerning the congruences for \(\sum _{k=0}^{(p-1)/2}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^3}{m^k}, \quad \sum _{k=0}^{p-1}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) ^2\left( {\begin{array}{c}4k\\ 2k\end{array}}\right) }{m^k} \quad \text {and}\quad \sum _{k=0}^{p-1}\frac{\left( {\begin{array}{c}2k\\ k\end{array}}\right) \left( {\begin{array}{c}3k\\ k\end{array}}\right) \left( {\begin{array}{c}6k\\ 3k\end{array}}\right) }{m^k}\) modulo \(p^3\) , where p is an odd prime representable by some suitable binary quadratic form and m is an integer not divisible by p.
