# Fermat Last Theorem: A proof for a special case

In summary, the speaker is asking for their proof of Fermat's Last Theorem to be taken seriously, and admits to a mistake in the figure. They then provide a new version of their proof and ask for feedback. They also receive advice on how to improve the presentation of their proof.
Please take me seriously, I know proving Fermat last theorem is not easy but I may have just proved it for a special case, everything is in the picture, so please tell if what I did is wrong or it's okay. I may have not reached the proof but I think I was on the way, just see the picture.
Thank you for your time

#### Attachments

• Fermat Last Theorem(Special Case Proof).jpg
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Please take me seriously, I know proving Fermat last theorem is not easy but I may have just proved it for a special case, everything is in the picture, so please tell if what I did is wrong or it's okay. I may have not reached the proof but I think I was on the way, just see the picture.
Thank you for your time

I see a mistake, you've written ##({b \over 2})^2## instead of ##{b^2 \over 2}##.

verty said:
I see a mistake, you've written ##({b \over 2})^2## instead of ##{b^2 \over 2}##.

Yeah you're right, I made this mistake in the figure, but I wrote ##{b^2 \over 2}## in the equations.

I can't read that line after "Simplifying 3", can you write out the line just before that and the line just after that here so I can see it? At the moment it looks wholly false to me.

The fact that it is not important enough for you to type it out rather than requiring others to read poorly photographed long-hand is enough to convince me that it is not important enough for me to try to read it!

It would help a lot if you at least precisely stated what it is you are trying to prove.

EDIT: And what your manipulations are intended to do. You come up with equation 3 and then seem to simplify it all the way down to bk = bk. What was the point of all of that?

Last edited:
A new decent version

I am sorry for not replying, this is a more decent version of what I was trying to show you guys. I think I got the theorem proven for any number b that is odd and where b=a+(a+n) but I don;t know how to continue when b is even.

#### Attachments

• Fermat Last Theorem Proof.pdf
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This proof lacks the usual standard of rigour, making it vague and hard to understand. You should be more precise about your intentions.

Too many boxes in the illustrations
The diagram at the beginning is difficult to read. Rather than having boxes (or "callouts") with a, b, and a + n, with curved lines going over to the two vertical lines, it would be better to have two horizontal lines of equal length, with one of them divided into two parts. Put a label of b on the undivided line, and put labels of a and a + n on the divided line. No boxes, no curved lines.

The second figure is even harder to understand, due to the boxes that represent the quantities (a + n)2, a2, and b2/2, as well as the other boxes that clutter up the drawing.

Equation numbering
Examples such as the one shown below confused me at first.
1 - b = a + (a + n)

As shown, it looked to me like the left side was 1 - b, not b as you intended.
The usual practice when equations are numbered is to put the number after the equation, in parentheses, like this:
b = a + (a + n) (1)

I wrote the parenthesized 1 in italics so as to not be interpreted as multiplication by 1.

Also, when equations are shown with numbers, usually only the more important ones are numbered, when they are discussed later in some detail. You shouldn't number steps that aren't equations, such as when you multiply both sides by bk-2.

Thank you h6ss and Mark44 I'll try working with your advice and make the proof more "elegant" and easy to read and understand.

## What is Fermat's Last Theorem?

Fermat's Last Theorem is a mathematical conjecture proposed by French mathematician Pierre de Fermat in the 17th century. It states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than 2.

## Who proved Fermat's Last Theorem?

The proof for Fermat's Last Theorem was first presented by English mathematician Andrew Wiles in 1994, after over 350 years of attempts by numerous mathematicians. However, Wiles' proof only covered a special case of the theorem, known as the Taniyama-Shimura-Weil conjecture, which was later proven by Wiles' former student Richard Taylor to be equivalent to Fermat's Last Theorem.

## What is the special case of Fermat's Last Theorem that was proven?

The special case of Fermat's Last Theorem that was proven by Andrew Wiles is when n = 4. This means that the equation a^4 + b^4 = c^4 has no integer solutions for positive values of a, b, and c. This was a significant breakthrough, as it was the first successful proof for any case of Fermat's Last Theorem.

## Why is Fermat's Last Theorem important?

Fermat's Last Theorem is considered one of the most famous and difficult problems in mathematics. Its proof has been pursued by mathematicians for centuries, and its solution has significant implications in number theory and algebraic geometry. It also serves as a reminder of the power and beauty of mathematics and the perseverance and creativity of mathematicians in solving complex problems.

## Is Fermat's Last Theorem fully proven now?

No, while Andrew Wiles' proof covered a special case of Fermat's Last Theorem, the full theorem is still not completely proven. The general case for all values of n greater than 2 is still open and remains an active area of research in mathematics.

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