Fermat's method of infinite descent is a proof method by contradiction. It aims to show that the integer solutions to an equation can be made smaller and smaller, but this is impossible because there are only finitely many natural numbers less than any given number.
The method was coined by the mathematician Pierre de Fermat, who is well known for the so called, "Fermat's last theorem", being one of the most well known maths problems in history, and remained unsolved for a century.
But aside from that, Fermat's method of infinite descent is a very useful technique in number theory, and has a wide range of applications.
Let's take an example from the following well-known problem of proving that the square root of 2 is irrational.
A number is said to be irrational if it doesn't belong to the set of rational numbers, and so it cannot be expressed as a fraction of two integers. Let's try to express the value of square root 2 as a fraction and see what happens.
This means that a must be even, as per the final equation. This means we can write a = 2c, where c is some positive integer smaller than a. Substituting it back into the last equation,
Once again, this means that b must be even. This means that we can write b = 2d, where d is clearly smaller than b. Substituting back into the equation we get,
Do you notice anything about this equation? It's identical to our first equation, but with a and b replaced by c and d. As we have discussed, we know that c and d are smaller than a and b, because we halved them.
Why did we do this though? It seems as though we worked in a circle and are back where we started with. However, the important difference, is that these new variables are smaller than the original variables. So for some a and b, which can be used as a fraction to represent the square root of 2, we can infinitely generate smaller numbers that also represent the square root of 2.
However, this is clearly impossible, as you can't have an infinite number of natural numbers which are less than some given value. So it is impossible for any such a and b to exist, and so the square root of 2 cannot be expressed as a fraction, and so must be irrational.
The key to these sorts of problems is to notice than you can create smaller solutions, and can then work to invoke Fermat's method of infinite descent.
Here is a somewhat harder problem that uses the same technique but requires an extra step. (If you need help, try googling quadratic residues, which we will cover shortly).
Are there any natural number solutions (so not zero) to the following equation?
For more explanations on Fermat's method of infinite descent, check out this website, https://en.wikipedia.org/wiki/Proof_by_infinite_descent#:~:text=In%20mathematics%2C%20a%20proof%20by,for%20a%20smaller%20number%2C%20leading. If you want to try your hand at some harder problems, have a go at some of these, https://brilliant.org/wiki/general-diophantine-equations-fermats-method-of/.
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