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Solutions of the Exponential Equation 7x^2 + 59y^2 = 3^m. A simple Algorithm producing all the primitive solutions
Roy Barbara
We provide a method, using essentially elementary arithmetic, to solve the exponential diophantine equation 7x^2 + 59y^2 = 3^m, which leads to a simple algorithm, with no use of ra…
The Pentagonal Inequality
Roy Barbara
Given a positive linear combination of five (respectively seven) cosines, where the angles are positive and sum to pi, the aim of this article is to express the sharp bound of the…
The Rational Distance Problem for Equilateral Triangles
Roy Barbara
Let (P) denote the problem of existence of a point in the plane of a given triangle T, that is at rational distance from all the vertices of T. In this article, we provide a comple…
Fermat's Last Theorem for the Exponent 3
Roy Barbara
`Fermat's Last Theorem for the exponent 3 has received numerous proofs, the most common of which being either in Euler's or in Gauss' style. This latter works entirely in the ring…
The Rational Distance Problem for Isosceles Triangles with one rational side
Roy Barbara, Antoine Karam
For a triangle , let (P) denote the problem of the existence of points in the plane of , that are at rational distance to the 3 vertices of . Answer to (P) is known to be…