An innovative integer distribution using Rotor-Router

In this paper, we study the operation of the Rotor-Router and attempt to nd a method, a process, thatallows integer grouping. We begin by dening the mapping f : N∗ −→ Z2, which associates to each integern ∈ N∗a position, f(n) ∈ Z2, in the plane, determined by the dynamics of the Rotor-Router. The mappingf thus provides a visual representation of the positive natural numbers in the plane. Filters are then appliedin order to retain only the integers possessing certain characteristics, such as divisibility, congruence, primality,etc. In other words, starting from n ∈ N∗, we select subsets satisfying properties such as d | n, n ≡ a(mod m), n prime, . . . The Rotor-Router mechanism produces an organization of integers whose observedregularity suggests an unexplored arithmetic distribution. In particular, the study of arithmetic progressions4k + 1 and 4k + 3, and more generally of congruence classes n ≡ a (mod b), has made it possible to highlightnon-uniform distribution phenomena, revealing the existence of remarkable arithmetic biases in the distributionof prime numbers.