The problems explore divisibility, systems of

Diophantine equations, functional equations, inequalities, trigonometry, chessboard problems, invariants, circles, angles, and lengths of line segments.

Lemma 2 lists all possible trigonometric

Diophantine equations with up to six.

The PSO + GI algorithm was tested on several

Diophantine equations and the results were compared to the standard PSO and SHC.

This contribution uses an algebraic approach based on general solutions of

Diophantine equations in [R.

Something like 900 years passed by between the time of Pythagoras and the time of Diophantus who worked with what is now called

Diophantine equations.

Sandor, Geometric theorems,

diophantine equations and arithmetic functions, New Mexico, 2002.

Such equations are called

Diophantine equations in the honour of Diophantus who studied them many centuries ago.

Other themes discussed during that academic year included: the Stolz--Cesaro, the intermediate and the mean value theorems, applications of complex numbers to polynomials,

Diophantine equations, and geometric probability.

Also it is known that the solution of sets of

Diophantine equations is a generator of primes.

In number theory many problems may be posed as

diophantine equations to be solved in integers.

In [1, 2] Quadratic

Diophantine equations with three unknowns have been considered for its parametric integral solutions.

In later years, students who stick with mathematics will learn more about

Diophantine equations in number theory courses.