**January 24**

Karel Hrbacek,
CUNY

**Representation of unlimited integers**

Nonstandard methods have been successfully applied to standard problems in number theory by R. Jin, T. Tao and others. A. Boudaoud and D. Bellaouar are pursuing the opposite direction: they are formulating number-theoretic problems in the language of nonstandard analysis and solving them by standard methods. Two examples of the kind of questions they consider are:

(1) Can every unlimited natural number n be represented in the form n = s + w_1w_2 where s is a limited integer and w_1, w_2 are unlimited?

(2) Can every unlimited natural number n be represented in the form n = w_1w_2 + w_3w_4 so that each ratio w_i / w_j is appreciable (ie, neither infinitesimal nor unlimited)?

I give a negative answer to question (1) (assuming Dickson’s Conjecture) and a positive answer to question (2).

A. Boudaoud, D. Bellaouar, Representation of integers: A nonclassical point of view, Journal of Logic & Analysis. 12:4 (2020) 1{31; K. Hrbacek, Journal of Logic & Analysis 12:5 (2020) 1–6.