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 \(2^{pq}\)


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