# Fermat's Theorem

**URL:** <https://boards.straightdope.com/t/fermats-theorem/582989>\
**Category:** Factual Questions\
**Created:** [May 24, 2011, 5:58am UTC](https://boards.straightdope.com/t/fermats-theorem/582989 "2011-05-24T05:58:23Z")\
**Posts on this page:** 20\
**Page:** 8

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**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\
**Post date:** [May 29, 2011, 11:33pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/141 "2011-05-29T23:33:28Z")

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> [@mla1](#):
>
> Antibob if you can not understand my claim please ask me which part you want to explain. When you do please take into account that I’m 16 and do not understand many (if any) complex functions. There are things you know that I cannot grasp as I have not been taught them. Also be specific about what part of the conjecture and how i should lay it out so that in future i don’t cause such great confusion.

Ok, I’ll play.

A complete statement of a conjecture means that in a single post you give all the details necessary so that a reasonably informed person (say with a modicum of mathematical knowledge) can read that one, single post and understand your conjecture.

From the several posts requesting clarification, it is obvious you have not yet met this standard.

For example, let’s say that all I wrote was:

```
x^n + y^n = z^n 

```

So far, this is YOUR style of posting. You give an equation but don’t explain your terms and simply let us flail about trying to understand how you are using the terms.

A complete statement would be:

The equation x^n + y^n = z^n, where x, y, z, and n are whole numbers, has no solutions for n \> 2.

Your last post states that the equation has solutions for p \> 0. Are there any other restrictions on p? By construction, obviously p \< n, so we have 0 \< p \< n. But you also use p = 0 when n = 2 and n = 3. So, which is it? p \>= 0? Or p \> 0? Also, clearly, if n \> 2, then p = n-2 will have no solutions (that’s Fermat’s last theorem, after all).

**ZenBeam** (and you should really try to get usernames correct - it’s just basic etiquette) has done an admirable job of trying to “interpret” your posts to make them understandable to the rest of us but still gets it wrong (at least according to you).

So, my suggestion is to take one of **ZenBeam** ’s posts and simple copy/paste. Edit it so that there are no longer incorrect statements. That would make for a very complete set of statements that we could use to understand your basic point.

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<div class="post-metadata">

**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 30, 2011, 12:36pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/142 "2011-05-30T12:36:20Z")

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n=2 p=0  
n=1 p=0  
Ok. I’ll do that. gove me time to take what he said and make it correct. I’l post it later today. Zedbeam thak you for attempting to get it.

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**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 30, 2011, 1:12pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/143 "2011-05-30T13:12:00Z")

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I’m fairly certain what mla1 is trying to say is:

x^n = a[sub]1[/sub]^n + a[sub]2[/sub]^n + … + a[sub]n-p[/sub]^n

has solutions for n, p, x, and all positive integers, n \>= 0, for 0=\< p \< n-1  
if n=2 p=0  
if n=1 p=0  
if n=3 p=0  
(i.e., p = 1 has no solutions either).

So

x^2 = a[sub]1[/sub]^2 + a[sub]2-0[/sub]^2 has solutions

x^3 = a[sub]1[/sub]^3 + a[sub]2[/sub]^3 does not have solutions.  
x^3 = a[sub]1[/sub]^3 + a[sub]2[/sub]^3 + a[sub]3-0[/sub]^3 has solutions.

x^4 = a[sub]1[/sub]^4 + a[sub]2[/sub]^4 does not have solutions.  
x^4 = a[sub]1[/sub]^4 + a[sub]2[/sub]^4 + a[sub]4-1[/sub]^4 has solutions.  
x^4 = a[sub]1[/sub]^4 + a[sub]2[/sub]^4 + a[sub]3[/sub]^4 + a[sub]4-0[/sub] has solutions

x^5 = a[sub]1[/sub]^5 + a[sub]2[/sub]^5 does not have solutions.  
x^5 = a[sub]1[/sub]^5 + a[sub]2[/sub]^5 + a[sub]3[/sub]^5 does not have solutions.  
x^5 = a[sub]1[/sub]^5 + a[sub]2[/sub]^5 + a[sub]3[/sub]^5 + a[sub]5-1[/sub]^5 has solutions.  
x^5 = a[sub]1[/sub]^5 + a[sub]2[/sub]^5 + a[sub]3[/sub]^5 + a[sub]4[/sub]^5 + a[sub]5-0[/sub]

x^6 = a[sub]1[/sub]^6 + a[sub]2[/sub]^6 does not have solutions.  
x^6 = a[sub]1[/sub]^6 + a[sub]2[/sub]^6 + a[sub]3[/sub]^6 does not have solutions.  
x^6 = a[sub]1[/sub]^6 + a[sub]2[/sub]^6 + a[sub]3[/sub]^6 + a[sub]4[/sub]^6 does not have solutions.  
x^6 = a[sub]1[/sub]^6 + a[sub]2[/sub]^6 + a[sub]3[/sub]^6 + a[sub]4[/sub]^6 + a[sub]5[/sub]^6 does not have solutions.  
x^6= a[sub]1[/sub]^6 + a[sub]2[/sub]^6 + a[sub]3[/sub]^6 + a[sub]4[/sub]^6 + a[sub]5[/sub]^6 + a[sub]6-0[/sub] has solutions.

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<div class="post-metadata">

**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\
**Post date:** [May 30, 2011, 3:58pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/144 "2011-05-30T15:58:15Z")

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Ok, I think I see where you are going now. I can see 2 possible ways this can go.

1. Simple existence statement. There exists at least one p, 0 \<=p \< n-1, such that the equation:

has solutions.

1. Beyond existence - for (1) above, the values of p for which there are solutions to the equation are ???

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<div class="post-metadata">

**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 30, 2011, 4:01pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/145 "2011-05-30T16:01:04Z")

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the first thing you said is right.

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<div class="post-metadata">

**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\
**Post date:** [May 30, 2011, 4:13pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/146 "2011-05-30T16:13:02Z")

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You included the n = 1 and n = 0 cases. n = 1 is trivial:

x^1 = a[sub]1[/sub]^1

But how does the n = 0 case work? The right hand side has no terms:

x^0 =

I guess you could say it’s 0, but then you’d get

x^0 = 0

which doesn’t have any solutions. You’d have to let p = -1 to get an a[sub]1[/sub] term.

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<div class="post-metadata">

**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 30, 2011, 4:15pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/147 "2011-05-30T16:15:04Z")

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yes it does x^0=a^0

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<div class="post-metadata">

**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [May 30, 2011, 4:19pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/148 "2011-05-30T16:19:20Z")

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> [@mla1](#):
>
> yes it does x^0=a^0

That has n = 0 and p = 1, which doesn’t jibe with **Great Antibob** ’s statement that you agreed with.

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<div class="post-metadata">

**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 30, 2011, 4:28pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/149 "2011-05-30T16:28:47Z")

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no he said pis equal to or greater than 0

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<div class="post-metadata">

**Author:** ![Bryan\_Ekers](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bryan_ekers/32/183_2.png) [@Bryan\_Ekers](https://boards.straightdope.com/u/Bryan_Ekers)\
**Post date:** [May 30, 2011, 4:41pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/150 "2011-05-30T16:41:28Z")

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> [@Hypnagogic\_Jerk](#):
>
> Sheesh, I never knew some people were that hostile to math.

When I was a kid…

… a six killed my dad.

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<div class="post-metadata">

**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\
**Post date:** [May 30, 2011, 4:53pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/151 "2011-05-30T16:53:56Z")

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> [@mla1](#):
>
> no he said pis equal to or greater than 0

It requires p = -1, not p = +1. The last term is a[sub]n-p[/sub]^n, so n-p has to be at least 1. Since n = 0, that requires p = -1.

ETA: Either way, the n = 0 and n = 1 cases are boring. Just wanted to clarify what I was saying.

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<div class="post-metadata">

**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\
**Post date:** [May 30, 2011, 4:57pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/152 "2011-05-30T16:57:22Z")

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> [@mla1](#):
>
> no he said pis equal to or greater than 0

And less than or equal to n - 1.

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<div class="post-metadata">

**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [May 30, 2011, 5:18pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/153 "2011-05-30T17:18:54Z")

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Quoth **Bryan Eckers** :

> [@](#):
>
> When I was a kid…
> 
> … a six killed my dad.

Well, you know, six is afraid of seven.  
Because 7 8 9.

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<div class="post-metadata">

**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 30, 2011, 5:36pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/154 "2011-05-30T17:36:55Z")

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Zedbeam x^0 =a[sub]1-0[/sub]^0

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<div class="post-metadata">

**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\
**Post date:** [May 30, 2011, 6:18pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/155 "2011-05-30T18:18:28Z")

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> [@mla1](#):
>
> Zedbeam x^0 =a[sub]1-0[/sub]^0

But a[sub]1-0[/sub] is a[sub]n-p[/sub], so that would be n = 1, p = 0, **not** n = 0, p = 1.

Or, to say it a different way, n-p = 1, and n = 0, so you have p = -1.

And it’s **ZenBeam** , as **Great Antibob** pointed out several posts ago.

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<div class="post-metadata">

**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 30, 2011, 6:22pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/156 "2011-05-30T18:22:22Z")

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never mind you know what Antibob wrote? can you prove it

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<div class="post-metadata">

**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\
**Post date:** [May 30, 2011, 6:31pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/157 "2011-05-30T18:31:05Z")

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Prove what? That when n-p = 1 and n = 0, then p = -1?

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<div class="post-metadata">

**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 31, 2011, 1:27pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/158 "2011-05-31T13:27:00Z")

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no the theorem

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<div class="post-metadata">

**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 31, 2011, 1:32pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/159 "2011-05-31T13:32:02Z")

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n-p does not always =1 it can =2 it can =1

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<div class="post-metadata">

**Author:** ![mla1](https://avatars.discourse-cdn.com/v4/letter/m/c0e974/32.png) [@mla1](https://boards.straightdope.com/u/mla1)\
**Post date:** [May 31, 2011, 1:34pm UTC](https://boards.straightdope.com/t/fermats-theorem/582989/160 "2011-05-31T13:34:21Z")

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If you guys are willing to try and prove it I will work on patterns and things that will make the proof easier but I need at least one person to try have a go. If you still don’t understand the formula don’t be afraid to ask and I’ll do my best to answer.

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