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DIOPHANTINE EQUATIONS IV - THE GOOD ORDER PRINCIPLE


Pierre de Fermat established a form of induction called the "Method of Infinite Descent." This method is used when we want to demonstrate that certain diophantine equations have no solution. Pierre de Fermat demonstrated the case no = 4 from Fermat's last theorem (UTF). In the infinite descent method we assume the existence of an integer and positive solution and from it we show that we can obtain another solution of integer and positive value smaller than the previous one. Proceeding in this way, we constructed an infinite decreasing sequence of positive values. However, the Good Order Principle states that every nonempty set of natural numbers has a smaller element, and thus we come to a contradiction. This contradiction stems from the assumption that the problem has a whole positive solution and so by the method of reduction to the absurd we conclude that the original problem has no solution.

Using the infinite descent method we observe that It has no integer solution other than trivial, (x, y, z), where x.y.z ≠ 0 and z> 0.

Suppose the positive integers x = x0, y = y0, z = z0 are a solution of with x0 and y0 cousins ​​among themselves. Notice that implies that , that is, It is a Pythagorean Terna. On the other hand, FIG. 10 and FIG.11 are prime to each other because if there was a prime p that divided and then p would divide x0 and y0, contrary to the fact that x0 and y0 are cousins ​​to each other. Therefore, It is a Primitive Pythagorean Terna. From this Primitive Pythagorean Terna we built a new Primitive Pythagorean Terna () such that > . Again, from the Early Pythagorean Terna () we built another Primitive Pythagorean Terna () such that > > . This process can be repeated indefinitely producing an infinite decreasing sequence of positive integers. > > . By the Principle of Good Order, a contradiction occurs. Therefore, we are forced to conclude that does not allow solution in the set of integers and positive numbers.

As an immediate corollary we get that the equation does not allow solution in the set of integers and positive numbers. In fact, if () were a whole positive solution of the equation , So () would be whole and positive solution of the equation contrary to the previous arguments. So, Fermat's last theorem (UTF) for the case no = 4 is true.

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