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# 数学代写|数论代写Number Theory代考|Germain’s Grand Plan

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## 数学代写|数论代写Number Theory代考|Germain’s Grand Plan

Here is the idea behind Germain’s ambitious “grand plan” to prove Fermat’s last theorem. She believed that for an odd prime $p$ that provided a counterexample to Fermat’s last theorem it would not be only a single prime $q$ that divides one of $x, y$, or $z$, as in Theorem 11.2, but that “the march of calculation indicates that there must be infinitely many (la marche du calcul indique qu’il doit s’entrouver une infinitê).”

She cites the case of $p=5$-that is, the case $x^5+y^5=z^5$ of Fermat’s last theorem-where the following primes would necessarily divide one of $x, y$, or $z$ :
$2 \cdot 5+1=11,2 \cdot 4 \cdot 5+1=41,2 \cdot 7 \cdot 5+1=71,2 \cdot 10 \cdot 5+1=101$, etc.
But, since it is impossible for infinitely many primes to divide just three numbers, this would mean no counterexample could exist for $p=5$.
Thus Germain’s plan was to provide a method that for each odd prime $p$ would produce infinitely many primes of the form $q=2 n p+1$ such that the primes $q$ would necessarily divide one of $x, y$, or $z$ for any given counterexample $x^p+y^p=z^p$ to Fermat’s last theorem.

The key in Theorem 11.2 was that for any power $x^p$ relatively prime to $2 p+1$ the power $x^p$ was congruent to either +1 or -1 modulo $2 p+1$. The key in Germain’s general method is the notion of consecutive nonzero powers modulo $2 n p+1$. Let’s look carefully at an example.

Consider $p=5$ and assume that $x^5+y^5=z^5$ is a counterexample to Fermat’s last theorem where $x, y$, and $z$ are positive integers. Further, we assume that $x, y$, and $z$ are relatively prime (otherwise we divide through by their greatest common divisor). What happens if a prime $q$, say 11 , does not divide any of $x, y$, or $z$ ? Then, since $x$ is relatively prime to 11 , we know by Theorem 6.1 that $x$ has an inverse modulo 11. Let $a$ be that inverse; that is, $a x \equiv 1$ (mod 11$)$. So, we can multiply the congruence $x^5+y^5 \equiv z^5(\bmod 11)$ through by $a^5$ to get $(a x)^5+(a y)^5 \equiv(a z)^5$ $(\bmod 11)$, which we can rewrite as $1 \equiv(a z)^5-(a y)^5$ (mod 11). Hence we conclude that modulo 11 the powers $(a z)^5$ and $(a y)^5$ are consecutive as residues in the complete system of residues $0,1,2, \ldots, 10$. Moreover, neither of these two powers can be congruent to 0 since 11 does not divide either $y$ or $z$ (and $a$ is relatively prime to 11 ).

We conclude that if 11 does not divide any of $x, y$, or $z$, then there must be two consecutive nonzero fifth powers modulo 11. But there aren’t! If we compute the fifth powers modulo 11 , we get only the following nonzero residues:
$$1^5 \equiv 3^5 \equiv 4^5 \equiv 5^5 \equiv 9^5 \equiv 1 \text { and } 2^5 \equiv 6^5 \equiv 7^5 \equiv 8^5 \equiv 10^5 \equiv 10 .$$

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

Sophie Germain was the first mathematician after Euler who made significant progress on Fermat’s last theorem. Fermat himself had proved the case $n=4$ using his method of infinite descent. In 1753, Euler proved the case $n=3$ (with a flaw that was later corrected by Legendre). Then, in 1825 , Legendre and Dirichlet independently proved the case $n=5$ basing their proofs on the work of Germain.

A dramatic episode in the long saga of Fermat’s last theorem occurred in the spring of 1847. Gabriel Lamé, who had proved the case $n=7$ eight years earlier, announced to the Paris Academy of Sciences at a meeting on March 1 that he had proved Fermat’s last theorem. Augustin-Louis Cauchy then announced that he too had a proof. However, both of these proofs came crashing down on May 24 when a letter arrived at the Academy from Ernst Eduard Kummer who pointed out that although unique factorization holds for the integers (see Theorem 3.4), it need not hold for number systems involving complex numbers such as those used in the two proofs in question. Lamé and Cauchy had simply assumed that unique factorization would still hold. Nonetheless, Kummer could verify Fermat’s last theorem for all primes less than 100 except for 37, 59, and 67 .

It would not be until the next century-and more than 350 years after Fermat wrote his famous note in the margin of the Arithmeticathat Fermat’s last theorem would finally be proved. Andrew Wiles announced a proof in the summer of 1993 in a series of lectures at Cambridge University. But a serious flaw in his proof was discovered in September and another year (and help from former student Richard Taylor) was needed for Wiles to fix his proof.

Wiles’s proof was in fact the final piece in a very complicated puzzle that began in the 1950s with a conjecture connecting topology and number theory called the Taniyama-Shimura conjecture. This conjecture was first shown to be related to Fermat’s last theorem by Gerhard Frey in 1984. The key breakthrough came in 1986 when Ken Ribet proved that the Taniyama-Shimura conjecture implies Fermat’s last theorem. Thus Wiles’s final assault on Fermat’s last theorem was his ultimately successful attack on a special case of the Taniyama-Shimura conjecture.

## 数学代写|数论代写Number Theory代考|Germain’s Grand Plan

$2 \cdot 5+1=11,2 \cdot 4 \cdot 5+1=41,2 \cdot 7 \cdot 5+1=71,2 \cdot 10 \cdot 5+1=101$等。

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