# 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 n0 has property P leads to the existence of a smaller positive integer n1<n0 that also satisfies P, then no positive integer has that property.

This is so because the reasoning that led from Since the process could be repeated thus leading to an infinitely

decreasing sequence of positive integers - which is impossible - the

assumption that

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

Any composite number is measured by some prime number.

If 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

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

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: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.

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 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.

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 agiven 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.

finitistic manner. Indeed, the method is based on the fact that every

subset of natural numbers has a least element. (The set

### 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

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