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The goal of this notice is to present a proof of Bachet's conjecture basedexclusively on the fundamental theorem of arithmetic. The novelty of this proofconsists in its introduction of a partial order on rational integers throughthe unique factorization property. In general, the proofs of Bachet'sconjecture by Lagrange - Euler's method assume necessary the use of infinitedescent. In the proposed proof we do not assume the existence of a "minimalsolution", but rather we show the existence of the desired solution through analgorithmic method.

An algorithmic proof of Bachet's conjecture and the Lagrange-Euler method