**March 7**

Bellaouar Djamel,
University 08 Mai 1945 Guelma

**Some generalizations on the representation of unlimited natural numbers**

Based on permanence principles of nonstandard analysis and as a continuation of the papers [1-3], we present some notes and questions on the representation of unlimited natural numbers. As a natural generalization, let $A$ be an unlimited $m$ by $n$ matrix with integer entries (i.e one of its integer entries is unlimited). Here we prove that every unlimited matrix $A$ with integer entries can be written as the sum of a limited matrix S with integer entries and the product of two unlimited matrices $W_1$ and $W_2$ with integer entries, that is, $A = S + W_1 \cdot W_2$. For further research, we propose several matrix representation forms.

Finally, we consider the numbers of the form $z = a+bi$ where $a$,$b$ are integers, which are called Gaussian integers. In the case when $a$ or $b$ is unlimited, the number $z = a+bi$ is said to be unlimited. Also, some notes on the representation of unlimited Gaussian integers are given.

[1] A. Boudaoud, La conjecture de Dickson et classes particulière d'entiers, Ann. Math. Blaise Pascal. 13 (2006), 103-109.

[2] A. Boudaoud and D. Bellaouar, Representation of integers: A nonclassical point of view, J. Log. Anal. 12:4 (2020) 1-31.

[3] K. Hrbacek, On Factoring of unlimited integers, J. Log. Anal. 12:5 (2020) 1-6.