数学代写|数论作业代写number theory代考|MATH2031

2022年10月12日

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数学代写|数论作业代写number theory代考|The Equation

Having just seen that the Diophantine equation $x^2+y^2=z^2$ has infinitely many primitive solutions (i.e., solutions in which $\operatorname{gcd}(x, y)=1$, it would be natural to expect that the equations $x^3+y^3=z^3, x^4+y^4=z^4$, and so on would also have infinitely many solutions. As we shall now see in this section and the next, this turns out to be not true, and in fact very strongly not true in the sense that we go from infinitely many solutions to no (nontrivial) solutions for each power $n>2$. This surprising fact was first conjectured in the 17-th century by Pierre de Fermat but was only proved in full generality about 350 years later in 1995 by Andrew Wiles. Wiles’ proof uses methods which are far beyond elementary number theory, but we shall see here that with some considerable effort we are able to use our elementary techniques to conclude that the Diophantine equation $x^4+y^4=z^4$ has no integer solutions beyond the “trivial” ones; i.e., the ones in which $x$ or $y$ is 0 . The exponent $n=4$ turns out to be easier to deal with than $n=3$ since with $n=4$, we are able, as we shall soon see, to make use of Theorem 9.2.

To get the desired result about $x^4+y^4=z^4$, we will first closely examine the related equation $x^4+y^4=z^2$, after which we can apply what we learned about it to achieve our main goal. So here is our central result:

Theorem 9.4. The only integral solutions of the Diophantine equation $x^4+y^4=z^2$ are the trivial solutions $\left{x=0, y, z=\pm y^2\right}$ and $\left{x, y=0, z=\pm x^2\right}$.

Proof. Our proof illustrates the method of “proof by descent” or “Fermat’s method of infinite descent,” as it is sometimes called. The idea is the following: Assume that the equation $x^4+y^4=$ $z^2$ has a non-trivial solution $\left{x_0, y_0, z_0\right}$ and then show that this assumption leads to another non-trivial solution $\left{x_1, y_1, z_1\right}$ with $z_1<z_0$. Repeating this process will then lead to a solution with a smaller $z_2$, and so on. Since starting with $z_0$ cannot produce an infinite strictly decreasing sequence of positive integers, our assumption that the original solution existed must have been false, and the theorem follows. So get ready for the tricky argument needed to produce the first new “smaller” solution $\left{x_1, y_1, z_1\right}$. In the course of this proof it will be necessary to introduce eight new positive integers: ${u, v}$ and later ${r, s}$ when applying Theorem $9.2 ;\left{z_1, t\right}$ and later $\left{x_1, y_1\right}$ when using the fact (already used once in the previous section) that if the product of two relatively prime integers is a perfect square, then each of the numbers must also be a perfect square.

数学代写|数论作业代写number theory代考|The Equation xn+yn=zn, n>2

Having now closely examined the specific Diophantine equations $x^2+y^2=z^2$ and $x^4+y^4=z^4$, and having reached totally different conclusions about the existence of integral solutions of them, we now turn to the general case: the equation $x^n+y^n=z^n$. Here is what Pierre de Fermat concluded about its solutions.

Conjecture 9.6. (Fermat’s Conjecture, 1637) If $n>2$, the Diophantine equation $x^n+y^n=z^n$ has no integral solutions except for the trivial solutions when one of $x$ or $y$ is 0 .

So Fermat stated that, even though $x^2+y^2=z^2$ has infinitely many integral solutions which we can explicitly describe (Theorem $9.2$ ), none of the corresponding equations with exponents greater than 2 have any non-trivial solutions. (We proved in the previous section, with considerable effort, that this is true of the equation $x^4+y^4=z^4$.) Fermat claimed to have a proof of his conjecture, but he never wrote his proof down, if indeed he really did have one. Though various special cases of the theorem were settled in the ensuing years (for example, in 1770 Leonhard Euler gave a proof for the case $n=3$ ), a general proof eluded mathematicians for centuries. Fermat’s Conjecture was finally resolved in 1995, when Andrew Wiles at Princeton University gave an extremely complicated proof requiring several hundred pages of very advanced mathematics. There remains hope that someone might discover an elementary proof of Fermat’s Conjecture (i.e., a proof using the kinds of arguments we used in the previous section to settle the matter for the case $x^4+y^4=z^4$ ), but this seems unlikely to occur because after more than 350 years no such elementary proof has been found.

We remark also that Fermat’s Conjecture is often referred to as “Fermat’s Last Theorem.” It has been given this name since Fermat had a habit of writing down conjectures but then not supplying the needed proofs. One by one these other conjectures were proved to be true by other mathematicians as the years passed, but no proof of Conjecture $9.6$ was found until 1995 . Hence it was, for many, many years, his “last theorem.”

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