Abstract
This work studies the natural powers of prime numbers as the building blocks of a Euclidian vector semi space. Some vectors generate the composite natural numbers by defining an appropriate geometrical norm. One also studies the structure of extended Mersenne numbers within this geometric point of view. Further geometric applications and extensions of the powers of natural numbers are also studied with the help of inward vector operations. Two research lines follow the first discussion on the geometrical aspects of natural numbers: the extension of the Fermat theorem and the Euler-Riemann function.
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