  In this book a multitude of Diophantine equations and their partial or
complete solutions are presented. How should we solve, for example, the
equation {\eta}({\pi}(x)) = {\pi}({\eta}(x)), where {\eta} is the Smarandache
function and {\pi} is Riemann function of counting the number of primes up to
x, in the set of natural numbers? If an analytical method is not available, an
idea would be to recall the empirical search for solutions. We establish a
domain of searching for the solutions and then we check all possible
situations, and of course we retain among them only those solutions that verify
our equation. In other words, we say that the equation does not have solutions
in the search domain, or the equation has n solutions in this domain. This mode
of solving is called partial resolution. Partially solving a Diophantine
equation may be a good start for a complete solving of the problem. The authors
have identified 62 Diophantine equations that impose such approach and they
partially solved them. For an efficient resolution it was necessarily that they
have constructed many useful tools for partially solving the Diophantine
equations into a reasonable time. The computer programs as tools were written
in Mathcad, because this is a good mathematical software where many
mathematical functions are implemented. Transposing the programs into another
computer language is facile, and such algorithms can be turned to account on
other calculation systems with various processors.
