Démonstration d'une propriété arithmétique récursive par récurrence binaire et prolongement quantique

We consider a recursive arithmetic property C defined on the set of natural numbers. We prove that C(n) is true for all n ∈ ℕ via induction on the binary size of integers. The proof combines classical induction for even integers and quantum induction for odd integers. The transition from bit to qubit allows extending C into a wave-like property C̃ in a complex Hilbert space of dimension 2k, where states associated with certain special integers |I> and |P> are orthogonal and entangled. The application of unitary transformations, particularly the Hadamard transformation, ensures the propagation of the property to all odd integers of a given size.