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In this paper, we have studied the third order variations on the Fibonacci universal code and we have displayed tables GH(3) a (n) we have defined for −20 ≤ a ≤ −2 and 1 ≤ n ≤ 100. Also, we have compared with the third order variations on the Fibonacci universal code and the second order variations on the Fibonacci universal code[2] in terms of cryptography and we have found that the third order variations on the Fibonacci universal code is more advantageous than the second order variations on the Fibonacci universal code.

In this study, we deal with a special form of the Brocard-Ramanujan equation, which is one of the interesting and still open problems of Diophantine analysis. We search for the positive integer solutions of the Brocard-Ramanujan equation for the case where the right-hand side is Mersenne numbers. By using the definition of Mersenne numbers, appropriate inequalities for the parameters of the equation, and the prime factorization of $n!$ we show that there is no positive integer solution to this equation. Thus, we obtain this interesting result demonstrating that the square of any Mersenne number can not be expressed as $n!+1$.

In this paper we did a generalization of Hadamard product of Fibonacci Q n matrix and Fibonacci Q −n matrix for continuous domain. We obtained Hadamard product of the golden matrices in the terms of the symmetrical hyperbolic Fibonacci functions and investigated some properties of Hadamard product of the golden matrices.