What Is Infinite Descent?

The method of Infinite Descent is so ubiquitous in Number Theory that rare is a book where the method is referred to in Index, let alone where the method is explicitly defined. As a pleasant exception, [J. Goldman, pp 13-14] and [Johnson & Richman, pp 13, 46] not only give a definition but also outline a short history of its usage.

Let \(P\) be a property that integers may or may not possess. If an assumption that a positive integer \(n_{0}\) has property \(P\) leads to the existence of a smaller positive integer \(n_{1}\lt n_{0}\) that also satisfies \(P,\space\) then no positive integer has that property.

This is so because the reasoning that led from \(n_0\) to \(n_1\) if applied to the latter will produce \(n_{2}\lt n_1\) that also has property \(P\). Since the process could be repeated thus leading to an infinitely decreasing sequence of positive integers - which is impossible - the assumption that \(P\) is satisfied by a positive integer implies a contradiction and, hence, is false.

Euclid makes use of the infinite descent in proving Elements VII.31:

Any composite number is measured by some prime number.

If \(A\) is composite it is divisible by \(B\lt A\). If \(B\) is prime we are done. If it's composite, it is divisible by \(C\lt B\), and so on. Then one of the two: either the so produced sequence of positive integers is finite and terminates with a prime factor of its predecessor (hence of its predecessor, and so on) and ultimately of \(A\), or the sequence is infinite - which would lead to a contradiction. Since the second possibility is impossible, the sequence is necessarily finite and VII.31 is proved.

For a complete and unambiguous association with the method of infinite descent, Euclid had probably to proceed thus: Assume \(P\) stands for the property of having only composite proper factors. If there is a positive integer \(A\), every factor of which is composite, let \(B\) be one of them. Since \(B\lt A\) and is composite, by the method of infinite descent, our assumption that there is an \(A\) is false: there are no integers without prime factors.

The method of infinite descent is commonly associated with the name of Pierre Fermat probably because he was the first to state it explicitly. Fermat never published a single work in number theory. There is just one result to which he supplied a complete proof:

The area of a right triangle whose sides have rational lengths cannot be the square of a rational number.

As was his wont, Fermat left a comment in the margin of his copy of Diophantus' Arithmetica:

This proposition, which is my own discovery, I have at length succeeded in proving, though not without much labor and hard thinking. I give proof here, as this method will enable extraordinary developments to be made in the theory of numbers.

Later on he specifies:

This is, however, impossible because there cannot be an infinite series of positive integers smaller any given integer we please. - The margin is too small to enable me to give the proof completely and with all details.

In a letter to Christian Huygens (with a reference to representing integers as sum of squares), Fermat clearly refers to the infinite descent as his own [Fauvel & Gray, p. 364]:

I have finally organized this according to my method and shown that if a given number is not of this nature there will be a smaller number which also is not, then a third less than the second, etc., to infinity, from which one infers that all numbers are of this nature.

As a matter of fact the method of infinite descent can be applied in a finitistic manner. Indeed, the method is based on the fact that every subset of natural numbers has a least element. (The set \(\mathbb{N}\) is well ordered.) Assume the set of all natural numbers that possess property \(P\) is not empty. Let \(n\) be the smallest integer in this set. Now, the existence of \(m \lt n\) that also has property \(P\) would contradict the selection of \(n\) as the least possible. Therefore, if the existence of an element with property \(P\) implies the existence of a smaller element, the set of all such elements ought to be empty.

References

W. H. Bussey, Fermat's Method of Infinite Descent, The American Mathematical Monthly, Vol. 25, No. 8 (Oct., 1918), pp. 333-337
J. Fauvel, J. Gray (eds), The History of Mathematics. A Reader, The Open University, 1987
J. R. Goldman, The Queen of Mathematics, A K Peters, 1998
B. L. Johnson, F. Richman, Numbers and Symmetry: An Introduction to Algebra, CRC Press, 1997

What Is Infinite Descent

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