Problems

The positive integers can be split into five distinct arithmetic progressions, as shown: \begin{align*} A&: \ \ 1, \ 6, \ 11, \ 16, \ ... \\ B&: \ \ 2, \ 7, \ 12, \ 17, \ ...\\ C&: \ \ 3, \ 8, \ 13, \ 18, \ ... \\ D&: \ \ 4, \ 9, \ 14, \ 19, \ ... \\ E&: \ \ 5, 10, \ 15, \ 20, \ ... \end{align*} Write down an expression for the value of the general term in each of the five progressions. Hence prove that the sum of any term in \(B\) and any term in \(C\) is a term in \(E\). Prove also that the square of every term in \(B\) is a term in \(D\). State and prove a similar claim about the square of every term in \(C\).

1. Prove that there are no positive integers \(x\) and \(y\) such that \[ x^2+5y=243\,723 \,. \]
2. Prove also that there are no positive integers \(x\) and \(y\) such that \[ x^4+2y^4=26\,081\,974 \,. \]