Dedicated to Carl F. Gauss.

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

Let there be two natural numbers ; SS and NN. such that : 2pS2^p | S and 2qN2^q | N , for arbitary whole numbers , pp and qq.

Let us define a new number MnM_n.

such that , MnM_n is the number formed by sum of first n+1n+1 and nn digits of SS and NN respectively.

Also SS and NN are jj and kk digit numbers , with provided that , (j,k)>n(j,k)>n , also , (p,q)<n(p,q) <n

Then evaluate the above summation modulo 2pq2^{pq}


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