In details

An infinite descent


The most famous Diophantine equation is the Fermat x equation.no + yno = zno. When n = 2 we have x² + y² = z² from where we get the Pythagorean suits. His solution appeared during classical antiquity in the work “The Elements” by the Greek mathematician Euclides. The next progress was made 1400 years later by Fermat, Leibniz and Euler. Since the 17th century, many of the greatest mathematicians have tried unsuccessfully to reconstruct the wonderful demonstration Fermat claimed to have for the fact that xno + yno = zno There is no solution for integers and positives when n> 2. Fermat said it didn't fit the margin of his copy of Diophantus's book "Arithmetica." It is reported that in 1742, the greatest mathematician of the 18th century, Leonhard Euler, asked his friend Clerot to search Fermat's house for some piece of paper with any indication of Fermat's demonstration, but nothing was found. However, Euler gave the first correct but incomplete demonstration for the case of exponent n = 3.

Note that if the exponent n> 2 is not a prime number, then the exponent is either a power of two or is divisible by some odd prime number p. In the first case, n = 4k and the equation can be rewritten as

(xk)4 + (yk)4 = (zk)4. However Fermat has shown that the sum of two fourth powers cannot result in a fourth power. In the second case n = pk, and the equation becomes (xk)P + (yk)P = (zk)P Therefore, to demonstrate that the equation has no solution to arbitrary integer powers, it is sufficient to demonstrate that the equation is not soluble when n = p, where p is an odd prime. We can simplify the problem even further if we observe that if x, y, z form a solution of Fermat's equation and any two of them are divisible by the same integer d, then d also divides the third (for example, if d divides xP and zPthen there are integers The and B such that xP = gives and zP = db; soon yP = zP - xP = db -gives = d(The - B), and so d is a divisor of yP. Therefore, it is sufficient to determine solutions that are relatively close two by two. They are called "primitive". If p is an odd prime then (-z)P = -zP and we can state Fermat's theorem as follows: “if p is an odd cousin, then xP + yP + zP = 0 has no integer solutions x, y, z that are relatively close two by two and such that xyz ≠ 0.

In case n = 4, the statement is assigned to Fermat. This demonstration is based on a form of induction he invented and called the "Infinite Descendant Method." This method has been successfully applied to numerous other problems and uses the indirect demonstration also known as the "Reductio ad Absurdum" demonstration. Thus the contradiction stems from the negation of the thesis and we conclude that the original thesis is true. The Descendant Method can be briefly described as follows: We assume that there is an integer and positive solution to a problem at hand, and from it we show that we can obtain another integer and positive solution that is smaller than the previous one and continue in this way. This argument is contradictory because if we start from a positive value and construct a decreasing sequence of positive values ​​from this given value, after a finite number of steps we get either zero or negative integers. So we come to a contradiction that stems from the assumption that the problem has a whole and positive solution, and thus, by reduction to the absurd, it follows that the problem has no solution. In the next column we will demonstrate the case n = 4 of Fermat's Theorem using the Descendant Method.

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