We will introduce a class of equations having roots in number theory and in algebraic geometry which are a generalization of both Fermat Last Theorem and Zariski surfaces.

First we will describe recent results obtained by Deligne, Oort and Mitsui.

Next we will present some ideas grown in duscusions with Schintzel, H. Iwaniec, Mycielski, Elkies and W. Domitrz concerning a specific equation of degree eight. Thus we will relate this equation to conjectures of Lang and Bombieri about integer points on surfaces of general type.

In the subsequent part of our talk we will present our reaserch based on ideas of Ulam and Grothendieck with its applications.

In the last part we will discuss Ulam Colatz Problem and presend results obtained in joint work with M. Zapala.