The beckman-quarles theorem for rational spaces

The Beckman-Quarles theorem states that every unit-preserving mapping from R ^d to itself is an isometry, for all d≥2. The analogues for the rational spaces Q ^d were established for all even dimensions, d,d≥6, as well as for all odd dimensions d of the form d=2n ²-1=m ², for integers n,m≥2. The purpose of this paper is to present a proof of the rational analogues of the Beckman-Quarles Theorem in dimensions d of the form d=2n ²-1, for all n3. The proof is also applicable in all the even dimensions d of the form d=4k(k+1), for k1, and in the real cases for all the dimensions d,d3.