I am going to do my project in the field of number theory. Number theory, a subject with a long and rich history, has become increasingly important because of its application to computer science and cryptography. The core topics of number theory are such as divisibility, highest common factor, primes, factorization, Diophantine equations and so on, among which I chose Diophantine equations as the specific topic I would like to go deep into.

Fermat's Last Theorem states that if n is a positive integer greater than 2, then the Diophantine equation x^n+y^n=z^n has no nontrivial solutions. Diophantine equation is an equation together with the restriction that the only solutions of the equation of interest are those belonging to a specified set, often the set of integers or the set of rational numbers. Fermat's last theorem has its origins in the mathematics of ancient Greece; two thousand years ago Pierre de Fermat constructed the problem in the form I described above. One great mathematician after another had been humbled by Fermat's legacy and for three hundred years no body had been able to solve it; I only had the potential to try two of them, where n=2 and 4, which I would discuss later in my project.

Fermat's last theorem was first introduced to me by my professor in a lecture as one of the world's unsolved mathematic problems. At that moment, as I could not understand the problem properly so I thought it was very simple. However, when I look at it now, I know why it is one of the most difficult problems in the world. The problem looks very straightforward because it based on the one piece of mathematics that everyone can remember â" Pythagoras' theorem: In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides. [pic]

This is the fundamental theorem that every innocent schoolchild is forced to learn. Now I am going to give a simple prove of this theorem here. The aim of the proof it to show that Pythagoras' theorem is true for all right-angled triangles. The triangle drawn below could be any triangle with the lengths of its three sides unspecified. Then put four of the same triangles together to make it a square as shown. The area of the large square can be calculated in two ways. 1. Measure the area of the large square as a whole. The area is therefore (x+y)*(x+y). 2. Measure the area of each element of the large square. The area of each triangle is (1/2)*x*y, and the area of the square in the middle is z^2, so the total area is z^2+4*(1/2)*x*y, which is 2xy+z^2 [pic]

Those two methods represent the same area, so we can write (x+ y)^2= 2xy+z^2. Rearrange the equation, it becomes x^2+ 2xy+ y^2= 2xy+ z^2, delete 2xy from both sides we get x^2+ y^2 = z^2, which is Pythagoras' theorem. The argument is based on the fact that the area of the large square must be the same no matter what method is used to calculate it. We then derive two expressions for the same area, make them equivalent, and eventually the inevitable conclusion is that the square on the hypotenuse, z^2, is equal to the sum of the squares on the other two sides, x^2+ y^2. This argument holds true for all right-angled triangles. The sides of the triangle x, y, z can represent sides of any right-angled triangles. [pic]

The importance of Fermat's Last Theorem

Why does Fermat's last theorem become so famous? One of the reasons is that it is so hard that even three centuries of effort failed to find a proof, and this led to its notoriety as the most demanding riddle in mathematics. However, this acknowledged difficulty does not necessarily mean that Fermat's Last Theorem is an important theorem; it seems that the provision of the theorem does not lead to any other mathematical successes. In my opinion, the importance of this theorem is during the process of proving it many other...

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