In this paper, we found that the Diophantine equation (p + 1) 2x + q y = z 2 has no non-negative integer solution where p is a Mersenne prime number which q − p = 2 and x, y and z are non-negative integers.

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Abstract:We apply an iterative sequence for finding the common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for tree inverse-strongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to finding solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of paper we utilize our results to study the zeros of the maximal monotone and some convergence problem for strictly pseudocontractive mappings.