Problem source

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 $C$.
\begin{questionparts}
\item Prove that there are no positive integers $x$ and $y$ such that
\[
x^2+5y=243\,723 \,.
\]
\item Prove also that there are no positive integers 
$x$ and $y$ such that
\[
x^4+2y^4=26\,081\,974 \,.
\]
\end{questionparts}