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# 数学代写|数论代写Number Theory代考|Waring’s Problem

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## 数学代写|数论代写Number Theory代考|Waring’s Problem

We have devoted a considerable amount of time in this book to problems involving sums of squares, in part because of the origins of these problems in the ancient mathematics of the Greeks, but also because of the intense interest in these problems shown by mathematicians such as Fermat, Euler, and Lagrange. Mostly we have done so, however, because the two main results concerning sums of squares-Fermat’s characterization of the numbers that can be written as a sum of two squares (Theorem 5.3 and its corollary) and Lagrange’s four squares theorem (Theorem 7.4)_are among the most deeply satisfying achievements in all of number theory.

A natural question to ask is whether there are any similar results for sums of other powers. For example, what numbers can be written as a sum of cubes? Fermat’s last theorem tells us that no cube is a sum of two cubes, but we saw in Problem 1.18 that it is possible for a cube to be a sum of three cubes. Might it be also be possible that there is a theorem analogous to Lagrange’s four squares theorem saying that for some small fixed number $c$, every number can be written as a sum of $c$ cubes?

In fact, it is fairly easy to guess what the number $c$ should be. To guess what the number $c$ should be for sums of cubes, let’s first try to see where the “four” comes from in Lagrange’s four square theorem. Since $8=2^2+2^2$ is less than $3^2$, it is clear that 7 , being 1 less than 8 , is going to require four squares: $7=2^2+1^2+1^2+1^2$. Then, roughly speaking, any number greater than 7 won’t ever need more than four squares because there will be more squares to work with. Now, for cubes, we can see that the number 23 plays a similar role. Since 23 is 1 less than $24=2^3+2^3+2^3$ and 24 is less than $3^3$ we can see that 23 is going to require nine cubes: $23=2^3+2^3+1^3+1^3+1^3+1^3+1^3+1^3+1^3$.

So, it might just be possible-and this is a pretty wild guess at this point-that every number $n$ can be written as a sum of nine cubes. Your confidence in this conjecture would increase substantially if you were to check that except for 23 every positive integer up to 238 can be written as a sum of fewer than nine cubes. Then, once again, 239 requires nine cubes: $239=4^3+4^3+3^3+3^3+3^3+3^3+1^3+1^3+1^3$.

In 1770 , Edward Waring, who was the sixth Lucasian Professor of Mathematics at Cambridge University from 1760 to 1798 (see Problem 3.37 for other Lucasian Professors), made the extraordinary claim that not only could every number be written as a sum of 9 cubes, but every number could also be written as a sum of 19 fourth powers, or as a sum of $37 \mathrm{fifth}$ powers, or more generally as a sum of $g(k) k$ th powers where
$$g(k)=\left\lfloor\left(\frac{3}{2}\right)^k\right\rfloor+2^k-2$$

## 数学代写|数论代写Number Theory代考|Fermat’s Last Theorem

We began this chapter on Euler and Lagrange with the letter of 1729 that Christian Goldbach wrote from Moscow to Euler that first opened for Euler the world of number theory. We conclude this chapter on the two great mathematicians of the eighteenth century with a letter that Euler wrote to Goldbach, almost twenty-five years later, in 1753, about the famous conjecture that we now call Fermat’s last theorem.
There’s another very lovely theorem in Fermat whose proof he says he has found. Namely, on being prompted by the problem in Diophantus, find two squares whose sum is a square, he says that it is impossible to find two cubes whose sum is a cube, and two fourth powers whose sum is a fourth power, and more generally that this formula $a^n+b^n=c^n$ is impossible when $n>2$. Now I have found valid proofs that $a^3+b^3 \neq c^3$ and $a^4+b^4 \neq c^4$, where $\neq$ denotes cannot equal. But the proofs in the two cases are so different from one another that I do not see any possibility of deriving a general proof from them that $a^n+b^n \neq c^n$ if $n>2$. Yet one sees quite clearly as if through a trellis that the larger $n$ is, the more impossible the formula must be. Meanwhile I still haven’t been able to prove that the sum of two fifth powers cannot be a fifth power. To all appearances the proof just depends on a brainwave, and until one has it all one’s thinking might as well be in vain.
Euler worked on the first case of Fermat’s last theorem-that is, the case $n=3$-in the years between 1753 and 1770 . His proof is not at all simple, and in fact at one particular point his proof was not even quite complete, and contained a flaw that was later corrected by Legendre. We will save our discussion of Euler’s proof of the case $n=3$ for later in the book, and for now mention only that Euler based his proof on the use of a complex number called a cube root of unity:

$$\rho=\frac{-1+i \sqrt{3}}{2}$$
where the number $\rho$ has the remarkable property that $\rho^3=1$ (that’s why $\rho$ is called a cube root of unity: because when you cube it, you get 1 , that is, unity).

Euler then worked with the number system consisting of complex numbers of the form $a+b \rho$ where $a$ and $b$ are integers. In particular, then, Fermat’s equation
$$x^3+y^3=z^3$$

can be rewritten as
$$(x+y)(x+\rho y)\left(x+\rho^2 y\right)=z^3$$
(see Problem 7.41).

## 数学代写|数论代写Number Theory代考|Waring’s Problem

1770年，爱德华·沃林(Edward Waring)——剑桥大学(1760年至1798年)的第六任卢卡斯数学教授(见问题3.37)——提出了一个不同寻常的主张:不仅每个数字都可以写成9个立方数的和，而且每个数字都可以写成19个四次方的和，或者$37 \mathrm{fifth}$次方的和，或者更一般地说，$g(k) k$次方的和
$$g(k)=\left\lfloor\left(\frac{3}{2}\right)^k\right\rfloor+2^k-2$$

## 数学代写|数论代写Number Theory代考|Fermat’s Last Theorem

$$\rho=\frac{-1+i \sqrt{3}}{2}$$

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