We introduce the Arithmetic Site: an algebraic geometric space deeply related to the non-commutative geometric approach to the Riemann Hypothesis. We prove that the non-commutative space quotient of the adèle class space of the field of rational numbers by the maximal compact subgroup of the idèle class group, which we had previously shown to yield the correct counting function to obtain the co...

In many practical situations, we know only the intervals which contain the actual (unknown) values of physical quantities. If we know the intervals x for a quantity x and y for another quantity y, then, for every arithmetic operation , the set of possible values of x y also forms an interval; the operations leading from x and y to this new interval are called interval arithmetic operations. For...

There is a well-known formula due to Jacobi for the number r2(n) of representations of the number n as the sum of two squares. This formula implies that the numbers r2(n) satisfy elegant arithmetic relations. Conversely, these arithmetic properties essentially imply Jacobi’s formula. So it is of interest to give direct proofs of these arithmetic relations, and this we do.

We propose a theoretical model to realize DNA made circuits based on in-vitro algorithms, to perform arithmetic and logical operations. The physical components of the resulting Arithmetic-Logic Unit are a variety of elements such as biochemical laboratories, test tubes and human operators. The advantage of the model is the possibility to perform arithmetic operations with huge binary numbers.

Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory. Thus he paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes a...

We propose a realizability interpretation of a system for quantier free arithmetic which is equivalent to the fragment of classical arithmetic without nested quantiers, called here EM1-arithmetic. We interpret classical proofs as interactive learning strategies, namely as processes going through several stages of knowledge and learning by interacting with the “environment” and with each other. ...