Intereting Posts

Some way to integrate $\sin(x^2)$?
Blowing up a singular point on a curve reduces its singular multiplicity by at least one
What IS conditional convergence?
How was the difference of the Fransén–Robinson constant and Euler's number found?
Why is the cross product of two vectors always orthogonal to the input vectors?
Software for generating Cayley graphs of $\mathbb Z_n$?
Computer software for solving mixed strategy Nash equilibrium
Equality of mixed partial derivatives
Calculating the limit of $^{1/n}$ as $n$ tends to $\infty$
Zero-divisors and units in $\mathbb Z_4$
Laplacian as a Fredholm operator
How can a random variable have random variance?
There is a unique polynomial interpolating $f$ and its derivatives
Efficient way to determine if a number is Perfect Square
Are numbers of the form $n^2+n+17$ always prime

In 1847, Lame gave a false proof of Fermat’s Last Theorem by assuming that $\mathbb{Z}[r]$ is a UFD where $r$ is a primitive $p$th root of unity.

The best description I’ve found is in the book *Fermat’s Last Theorem A Genetic Introduction to Algebraic Number Theory*. For the equation $x^n + y^n = z^n$, it says

That is, he planned to show that if $x$ and $y$ are such that the factors $x+y, x+ry, \dots, x+r^{n-1}y$ are relatively prime then $x^n + y^n = z^n$ implies that each of the factors $x+y, x+ry, \dots$ must itself be an $n$th power and to derive from this an impossible infinite descent. If $x+y, x+ry, \dots$ are not relatively prime he planned to show that there is a factor $m$ common to all of them so that $(x+y)/m, (x+ry)/m, \dots, (x+r^{n-1}y)/m$ are relatively prime and to apply a similar argument in this case as well.

- Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.
- Show that $\langle 13 \rangle$ is a prime ideal in $\mathbb{Z}$
- For which $d$ is $\mathbb Z$ a principal ideal domain?
- What is the index of the $p$-th power of $\mathbb Q_p^\times$ in $\mathbb Q_p^\times$
- What do we know about the class group of cyclotomic fields over $\mathbb{Q}$?
- How to prove that there exist infinitely many integer solutions to the equation $x^2-ny^2=1$ without Algebraic Number Theory

Nowhere else can I find any more detail as to how the rest of the argument goes. I don’t see how to use infinite descent here, can anyone fill in the details?

I would like to also mention Conrad’s great notes on Kummer’s proof for regular primes. While this does indeed give a proof, it seems quite sophisticated and I am hoping there is an easier method when one assumes (falsely, in general) that $\mathbb{Z}[r]$ is a UFD.

- Minimal polynomial of an algebraic number expressed in terms of another algebraic number
- Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers
- Find a formula for all the points on the hyperbola $x^2 - y^2 = 1$? whose coordinates are rational numbers.
- Where does TREE(n) sit on the Fast Growing Hierarchy?
- Is there any real number except 1 which is equal to its own irrationality measure?
- Are 14 and 21 the only “interesting” numbers?
- Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$
- Prime factor of $A=14^7+14^2+1$
- Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$
- Smallest number $N$, such that $\sqrt{N}-\lfloor\sqrt{N}\rfloor$ has a given continued fraction sequence

The paper (in French) to which Edwards refers in his book may be found here:

http://gallica.bnf.fr/ark:/12148/bpt6k29812/f310.image

My French isn’t good enough, however, to elaborate on the argument given above.

- Spaces with the property: Uniformly continuous equals continuous
- Uncountable union of multiples of measurable sets.
- What is the sum of this? $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$
- Optimisation Problem on Cone
- What are the theorems of mathematics proved by a computer so far?
- “Support function of a set” and supremum question.
- What is the best way to show that the exponential sequence doesn't uniformally converge on an unbounded interval
- If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$
- What is an intuitive meaning of genus?
- Classifying Types of Paradoxes: Liar's Paradox, Et Alia
- Converge or diverge? : $\sum_{n=1}^{\infty}\frac{\tan{n}}{2^{n}}$
- Equal elements vs isomorphic elements in a preoder
- Next step to take to reach the contradiction?
- How to compute the area of the shadow?
- Degree sequence of connected graphs