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数学代写|数论代写Number Theory代考|Primitive Roots

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Our goal-as it was for much of Euler’s life-is a proof of Lagrange’s famous theorem that every number can be written as the sum of four squares. As is often the case, in order to move toward this goal, we need to introduce a new concept. The notion of quadratic residues is a topic we will discuss in great detail in the next chapter, but in this section we will see how Euler made use of basic properties of quadratic residues in his unrelenting attempt to prove the four squares theorem.

For a given prime $p$, a number $a$ that is relatively prime to $p$ is called a quadratic residue of $p$ if it is congruent to a square modulo $p$; otherwise, $a$ is called a quadratic nonresidue. For example, 6 is a quadratic residue of the prime 19 , because $5^2=25 \equiv 6$ (mod 19$)$ ); that is, 6 is a “square” modulo 19.

For any odd prime $p$, only half of the numbers $1,2, \ldots, p-1$ will be quadratic residues, and the other half will be quadratic nonresidues. There are two ways to see this.

The most natural way to see this is that the quadratic residues are given by
$$1^2, 2^2, \ldots,\left(\frac{p-1}{2}\right)^2$$
The next squares, $\left(\frac{p+1}{2}\right)^2,\left(\frac{p+3}{2}\right)^2, \ldots,(p-1)^2$, just repeat these same numbers all over again, since
$$\frac{p+1}{2} \equiv-\frac{p-1}{2}, \quad \frac{p+3}{2} \equiv-\frac{p-3}{2}, \ldots, p-1 \equiv-1(\bmod p)$$
so, among $1,2, \ldots, p-1$ there are $\frac{p-1}{2}$ quadratic residues and, therefore, $\frac{p-1}{2}$ quadratic nonresidues.

For example, we see above that $14^2=196 \equiv 6(\bmod 19)$, which just repeats the quadratic residue 6 already produced by $5^2$; this is because $14=19-5 \equiv-5(\bmod 19)$.

数学代写|数论代写Number Theory代考|Lagrange

It is almost inconceivable to us now, from our perspective in the twentyfirst century, at a time when travel is so easy, to realize that the two great mathematicians of the eighteenth century, Euler and Lagrange, never met one another. Yet their lives and their mathematics were deeply intertwined. We tend to think of Lagrange as a French mathematicianand the French certainly do-but he was born in Turin as Giuseppe Lodovico Lagrangia. And at the age of nineteen he was appointed as professor of mathematics at the Royal Artillery School in Turin.

In that same year, 1755, Lagrange wrote for the first time to Euler-his elder by almost thirty years, and who was by then at the Royal Academy in Berlin-about methods he had developed for solving problems in an area we now call the calculus of variations involving curves such as the tautochrone, a curve with the remarkable property that if two point masses begin falling simultaneously from two points anywhere on the curve, and fall along the curve only under the influence of gravity, then they will reach a given fixed point at exactly the same time. Euler was tremendously impressed by the young Lagrange and in 1756, even arranged for him to be offered a position in Berlin, but Lagrange declined. However, when Euler left Berlin in 1766 to return to St. Petersburg, it was Lagrange who took his place, and he was to remain there for twenty years as director of mathematics, before eventually moving to Paris.

We have already mentioned many of Lagrange’s major contributions in number theory: his proof of Wilson’s theorem, Theorem 6.3; his theorem on the number of solutions of a polynomial congruence, Theorem 6.5; and in the next section we will see his crowning achievement in number theory, his four squares theorem, Theorem 7.4. But Lagrange also made vital contributions across the entire field of mathematics: he invented the method known as variation of parameters for solving differential equations; it perhaps goes without saying that the method of Lagrange multipliers is due to him; while in Berlin he wrote his great treatise Mécanique Analytique on mechanics; and one of the most fundamental theorems in group theory is called Lagrange’s theorem, which says that the order of any subgroup divides the order of the group (thus generalizing the important corollary to Fermat’s little theorem about the order of $a$ dividing $p-1$ ).

数论代写

$$1^2, 2^2, \ldots,\left(\frac{p-1}{2}\right)^2$$

$$\frac{p+1}{2} \equiv-\frac{p-1}{2}, \quad \frac{p+3}{2} \equiv-\frac{p-3}{2}, \ldots, p-1 \equiv-1(\bmod p)$$

MATLAB代写

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