One of the interesting aspects of arithmetic is that mathematical proofs can be constructed without needing a large theoretical arsenal. These proofs are supported by reasoning of a certain subtlety, playing with the notions of infinity and the absurd, and hence non-trivial results can be obtained. This reasoning is easily accessible intuitively because it relates to the integers, giving arithmetic a specific formative character to students undergoing their apprenticeship in proof.
The history of mathematics offers us a large choice of proofs, some more formal, some less, some further from intuition, some closer. We have, moreover, commentaries by mathematicians regarding the elegance or the rigour of certain of these proofs, to which we can refer.
The corpus of texts we have chosen for reading revolves around « Fermat’s Little Theorem » which is part of the final programme in secondary school. The basic theoretical baggage is then limited to a single property which appears in different forms – Euclid’s Lemma, Gauss’Theorem, The Fundamental Theorem of Arithmetic – according to one’s point of view and to the context. The essential core of these methods of
proof also manifests itself in different forms (infinite descent, the principle of recursion, the use of the smallest integer in a set of integers).