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.

 A. Boudaoud, La conjecture de Dickson et classes particulière d'entiers, Ann. Math. Blaise Pascal. 13 (2006), 103-109.
 A. Boudaoud and D. Bellaouar, Representation of integers: A nonclassical point of view, J. Log. Anal. 12:4 (2020) 1-31.
 K. Hrbacek, On Factoring of unlimited integers, J. Log. Anal. 12:5 (2020) 1-6.

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