# 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 $$2^{pq}$$

×