> with(plottools):
f :=hexahedron([-50,0,0],4),hexahedron([-40,0,0],3), hexahedron([-30,0,0],2),hexahedron([-20,0,0],1.5),hexahedron([-10,0,0],1), hexahedron([0,0,0],0.75), hexahedron([10,0,0],0.5),hexahedron([20,0,0],0.25),hexahedron([30,0,0],0.15),hexahedron([40,0,0],0.05):
plots[display](f,style=patch);

Fermat and His Method of Infinite Descent

Pierre de Fermat (1601 - 1665) is quite famous for his contributions to mathematics even though he was only considered an amatuer mathematician. Fermat received his degree in Civil Law at the University of Orleans before 1631 and served as a lawyer and then a councillor at Toulouse. The political and legal path Fermat chose is attributed to the trends of upward social mobilty during this time. In other words, Fermat chose a career that would lead to politcal and financial power. Although his career did not directly revolve around mathematics, it seems apparent that his interests did. Fermat was known by the greatest mathematicians of his time and most had very distinct opinions of him. While Pascal thought of him as "the greatest mathematician in all of Europe," Descartes called him a "braggart," and Wallis called him "that damned Frenchman." Whatever opinions people had of him, Fermat proved to have a monumental impact on the world of mathematics through his findings and methods.

Fermat was one of the first mathematicians to use a method of proof called the "infinite descent." In his words, he utilized this method "because the ordinary methods now in the books were insufficient for demonstrating such difficult propositions." At first, Fermat found this method only useful for proving negative assertions such as "there is no right angled triangle in numbers whose area is a square." Although proving affirmative assertions with this method is more difficult, Fermat realized how useful this method could be in proving those difficult propositions that had remained unproven for many, many years. The basic method of the infinite descent is as follows: Assume one wants to prove no solution exists with a certain property. First, assume a positive integer, x, posseses such a property. Next, deduce that there exists some positive integer y < x which also has the same property. Repeat this argument an infinite amount of times thus infinitely descending through all integers. This contradicts the fact that there must be a smallest positive integer with this property. Therefore, no positive integer exists with the proposed property.

First, a simple example will illustrate this method. Let's prove that the (sqrt 2) is irrational. First, assume (sqrt 2) has a rational solution. Therefore, (sqrt 2) can be represented as the quotient of two positive integers as follows: (sqrt 2) = a1/b1 where a1 > b1. Using the following equality,

1/(sqrt(2) - 1) = (sqrt(2) + 1)/1;

replace the (sqrt 2) on the left-hand side of the equation with a1/b1. Solving for (sqrt 2), the following equation is derived:

> sqrt(2) = (2 - a1/b1)/(a1/b1 -1);

Simplified, this becomes

> sqrt(2) = (2*b1 - a1)/(a1 -b1);

Using the fact that 1 < a1/b1 <2, both the numerator and denominator are positive integers. Therefore, the numerator can be represented as a2 and the denominator as b2 where, again, a2 > b2. Clearly, a2 < a1 and b2 < b1 since a1 > b1. So, (sqrt 2) is now the following:

> sqrt(2) = a2/b2;

>

This process can be repeated over and over an infinite amount of times, each time a(n+1) is less than a(n) and b(n+1) is less than b(n). In other words, there is an infinite descent through all positive integers which satisfy (sqrt 2) = a/b. Therefore, there exist no smallest positive integers, a and b, that satisifty (sqrt 2) = a/b. This is a contradiction, and thus, (sqrt 2) must not be a quotient of positive integers and therefore is irrational.

Fermat effectively used this method of infinite descent in his circa 1640 where he proved that, the area of a pythagorean triangle cannot be a square. Or, in equation form, there is no integer solution to the following:

> (x^4) + (y^4) = (z^2);

First, he assumed that there did exist such an integer solution that satisfies this equation. The equation can then be rewritten as follows:

> (x^2)^2 + (y^2)^2 = z^2;

So, (x^2), (y^2), and (z^2) are pythagorean triples. Therefore, each can be rewritten as follows:

> y^2 = 2*p*q;

> x^2 = p^2 - q^2;

> z = p^2 + q^2;

Since 2pq is a square, either p or q is even. So, (p^2) can be rewritten as a pythagorean triple.

> x^2 + q^2 = p^2 ;

Here, x, q, and p are the pythagorean triples and can be rewritten as follows:

> q = 2*r*s;

> x = r^2 - s^2;

> p = r^2 + s^2;

Since 2pq is a square (y^2), q can be set equal to 2(u^2) and p can be set equal to (v^2).

> q = 2*u^2;

> p = v^2;

> 2*u^2 = 2*r*s;

Since this is true, r can be set equal to (g^2), and s can be set equal to (h^2).

> r=g^2;

> s=h^2;

Using this r and s, and using p = (v^2), the following equation is derived:

> v^2 = g^4 + h^4;

Notice, this is of the same form as the original proposition, but v < z. This process can be repeated an infinite number of times, each time yielding smaller integer solutions. This is a contradiction since there must be a smallest solution. Therefore, no positive integer solutions exist which satisify the original proposition. In Fermat's words, "if the area of such a triangle were a square, then there would also be a smaller one with the same property, and so on, which is impossible."

One must be careful when using this method of proof to state the "specific reduction process" in order to make the proof valid. This mistake is clearly illustrated in Fermat's letter to Huygens where he gives a proof to the positive assertion that every prime number of the form 4k + 1 is the sum of two squares. First he makes the assumption that there exists some prime of the form 4k + 1 which is not composed of two squares. Then, he states that there must exist another smaller prime of the same form which is not composed of two squares. Repeating this process, the prime integer of the form 4k + 1 descends until it becomes the smallest such integer of that form, 5. Therefore, 5 should not be the sum of two squares. However, 5 = (2^2) + (1^2) which contradicts the original assumption. Therefore, every prime number of the form 4k + 1 is composed of two squares. The problem with this proof is that Fermat did not include the specific reduction process whereby if there exists a prime of the form 4k + 1 that is not the sum of two squares, then there exists a smaller such prime with those same properties. Did Fermat just not include it out of carelessness? Did Fermat even know this reduction process to be absolutely true? It turns out that Fermat's conjecture was, in fact, true. Euler asserted this to be true by saying, "For any N = a^2 + b^2, let q = x^2 + y^2 be a prime divisor of N. Then N/q has a representation u^2 + v^2 such that the representation N = a^2 + b^2 is on of those derived by composition from it and from q = x^2 + y^2." This is one route that reduces the decomposition of p into a smaller prime with the same properties. This is the step Fermat left out that completes the proof.

Fermat based the method of infinite descent on many of the challenging problems he asserted such as: No cube can be the sum of two cubes; all square powers of 2 increased by one are prime; (x^2) + 2 = (y^3) has only one solution; and 2(x^2) - 1 = p, 2(y^2) - 1 = (p^2) have only integral solutions for p = 1 and p = 7. Two very famous assertions were that (2^2^n) + 1 = a prime number and (x^n) + (y^n) = (z^n) has no integral solutions for n > 2. The latter assertion is knows as Fermat's "Last Thereom." Fermat published no known proof to his "Last Theorem" but did prove two cases, when n = 3 and n = 4. Fermat actually used the before stated assertion, (x^4) + (y^4) = (z^2) has no integer solution to conclude that there is no integer solutions for (x^4) + (y^4) = (z^4). In a completely dissimilar way, he proved that (x^3) + (y^3) = (z^3) has no integer solution. He did not prove the general case but mearly stated "which this margin is too narrow to contain." Some believe that Fermat assumed that the method of infinite descent would work for all cases if it worked for those two. This is a careless assumption. Others believe he kept the general proof to himself not wanting to share this incredible accomplishment with anyone else. Fermat also made a careless assertion when he stated that (2^2^n) + 1 = a prime number. In fact, Euler dissproved this theorem with the case of n = 5.

> x = 2^(2^5) + 1;

> y = 4294967297/641;

So, Fermat's assertion was false which, indicated that he had too much faith in his method of infinite descent. He stated in August of 1640, "I don't have an exact proof of this, but I've excluded such a large number of divisors by infallible proofs, and I have such great intuitions(?), that I would be at great pains to retract [the conjecture?]" Although Fermat's intuition here was wrong, this was a rare case. Most mathematicians accepted Fermat's intuition of his "Last Theorem" as truth and have actively sought its proof. In fact, rewards have been offered throughout time for this proof. This problem has inspired many great mathematicians much the way the construction problems of the ancients did. In 1994, Andrew Wiles published what is believed to be the proof of Fermat's "Last Theorem." The breakthrough was a link discovered by Ken Ribet at the University of California at Berkeley to Fermat's Last Theorem with another unsolved problem, the Taniyama-Shimura conjecture. This led to the 150 page proof by Andrew Wiles which was finalized in 1994. Fermat's contributions to mathematics remain a hot topic even today. It is safe to say that he had a tremendous impact on the mathematical world, especially for what many considered an "amatuer" mathematician.

Sources:

The Mathematical Career of Pierre de Fermat (1601- 1665) , Mahoney, Michael Sean Princeton University Press, 1994.

A History of Mathematics , Boyer, Carl B John Wiley & Sons, Inc., 1991.

Elementary Introduction to Number Theory , Long, Calvin T. Waveland Press, Inc., 1995.

http://web-cr02.pbs.org/wgbh/nova/proof/wiles.html, "Solving Fermat: Andrew Wiles"