Fermat’s “infinite descent”


Yves Tanguy, “Indefinite Divisibility” (1942)

In a letter to his friend Pierre de Carcavi, dated August 14, 1659, Pierre de Fermat (1601 – 1665) announces a method of mathematical proof, which he names “descent infinie ou indéfinie” (infinite or indefinite descent). It is a method similar to the mathematical induction in that both methods establish an infinite sequence of statements in which every statement necessarily validates the truth of the next one, and therefore the truth of all statements collectively is ensured by this domino – like process. The mathematical induction is an “infinite ascent” method, as each next statement of the domino is created by increasing a positive integer index. The infinite descent creates a similar domino in which each next statement of the domino is created by decreasing a positive integer index, a process which inevitably is at some point blocked by the well ordering principle (the fact that every non-empty set of positive integers contains a smallest number). Though it is possible to move upwards on the “ladder” of positive integers indefinitely, as happens in mathematical induction, it is not similarly possible when moving downwards, as there exists a least positive integer and the descent cannot be indefinite. Therefore the process arrives at a contradiction: any proposition that leads to an indefinite descent, within the positive integers, is necessarily false. As Fermat noted in his letter, it is a method particularly suited for proving negative propositions. He briefly discusses in his letter how the infinite descent could be applied to prove that “there is no right triangle with integer side lengths whose area is equal to the square of an integer“. Assuming that such a triangle exists, then one can prove that there necessarily exists a second, smaller triangle with the same property and then, by applying the same arguments to that, a third, a fourth and so forth ad infinitum. But an infinite sequence of positive integers less than a given integer is impossible, from which Fermat concludes that a contradiction is established, proving that a triangle possessing the specific property does not exist (a detailed description of the proof can be found here). The above proposition is equivalent to the statement “there are no three positive integers a, b, c such that a4 + b4 = c4“, which is a special case of the notorious Fermat’s last theorem (there are no three positive integers a, b, c such as an + bn = cn , for any integer n greater than 2) for n=4. In his letter to de Carcavi, Fermat describes a number of other problems that could be answered by means of his infinite descent method, including:

“Every prime of the form 4k+1 is equal to the sum of two squares”

“Every positive integer is either a square or the sum of two or three or four squares”

Though Fermat announces the infinite descent as a method of his own invention, the core idea was known since antiquity. In Book 7 of Euclid’s “Elements”, proposition 31 states that “any composite number is measured by some prime number” (where “a is measured by b” means that b is a factor of a) and the corresponding proof involves an infinite descent argument (the proof can be found here).