Dedicated to Carl F. Gauss.

$\sum(M_n +M_{n+1})^2$

Let there be two natural numbers ; $S$ and $N$ . such that : $2^p | S$ and $2^q | N$ , for arbitary whole numbers , $p$ and $q$ .

Let us define a new number $M_n$ .

such that , $M_n$ is the number formed by sum of first $n+1$ and $n$ digits of $S$ and $N$ respectively.

Also $S$ and $N$ are $j$ and $k$ digit numbers , with provided that , $(j,k)>n$ , also , $(p,q) <n$

Then evaluate the above summation modulo $2^{pq}$