Geometry and Diophantine equations have been ever-present in mathematics. Diophantus of Alexandria was born in the 3rd century (as far as we know), but a systematic mathematical study of word equations began only in the 20th century. So, the title of the present article does not seem to be justified at all. However, a linear Diophantine equation can be viewed as a special case of a system of wo...

We show how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equation G, = wp" ... p', (where (G,, }I='o is a binary recurrence sequence with positive discriminant), for arbitrary values of the parameters. We apply this to the equation x2 + k = ... ps', which is a generalization of the Ramanujan-Nagell equation x2 + 7 = 2Z. We ...

We give a new proof of the validity of Cornacchia’s algorithm for finding the primitive solutions (u, v) of the diophantine equation u + dv = m, where d and m are two coprime integers. This proof relies on diophantine approximation and an algorithmic solution of Thue’s problem.

DIOPHANTINE EQUATIONS INVOLVING ARITHMETIC FUNCTIONS OF FACTORIALS Daniel M. Baczkowski We examine and classify the solutions to certain Diophantine equations involving factorials and some well known arithmetic functions. F. Luca has showed that there are finitely many solutions to the equation:

In this paper, we discuss variations on the BrocardRamanujan Diophantine equation.

Let τ(n) be the Ramanujan τ -function. We prove that for any integer N the diophantine equation 74000