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May 30th, 2013, 11:57 AM   #1
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A new approach to Fermat's last theorem

In this topic I will try to prove the first case of Fermat's last theorem (FLT1 as it is often written), using only high school maths, well known at Fermat's time, with one only exception. All numbers involved are positive integers and N is a prime. We'll need the following sentences:

1. Fermat's little theorem:

is divisible by N. Or

1b. So if

and because


If N is an odd prime then we also have:
and the + or - signs in the left side match the respective + or - signs in the right side
If N is an odd prime, we also have

If N is an odd prime, we also have

3?. The reverse also holds:

Every prime greater than 3 can be written in the form 6q1.

5. r\:N|b" />.
This holds for more than two factors a,b

6. If the product of two coprime numbers is the pth power of a number, then each of those coprime numbers is also the pth power of a number:


So the goal is to prove that if N divides none of X,Y,Z and N>2 then we can't have

It is obvious that we will assume:

Step 1. Basic relations between X,Y,Z,N

We first show that

Suppose this wasn't true. Then there would be a prime Q common in these two numbers:

and if

and as Q is a prime, we get j=Q and so

but since

we conclude that

which is false, because X and Y are coprimes. So:

and from (1) we see that

is all "hidden" in N:

and as N is a prime:

which is false. Therefore:

and since:

by sentence 6 we end to:
N,R)=1" />
From Fermat's little theorem:

and as N doesn't divide R, by sentence 5 it follows
sentence 2a)\:N|D-1\rightarrow\sentence\:3)N^2|D^N-1=P_{XY}-1" /> (1)
On the other hand
sentence 2b) Z^N-N^2T\rightarrow N^2|Z^N-X-Y" />

Combining these last three equalities, we get:

Step 2 .
Consider the three Euclidean divisions

with the three (positive) remainders a,b,c are smaller than N. Then:
r\:0." />
By sentence 4, if N>3 (the case N=3 has been proved long ago), we consider two cases
Case I. N=6q+1
Let us assume the following Euclidean division (m<N)


and we have two subcases here:

and so:

Given that N can't divide a (or we would have a=0) and N can't divide q since q<N, then by sentence 5 what remains is:

Here we have:
and then

and likewise we now yield:

In a way similar to the one we used in subcase Ia), we end to:

and the conclusion is the same:

Of course


In the same way, we show

Case ??. N=6q-1.
Considering the same Euclidean division

we conclude in an analogous way that:

and we split it in two subcases again
???1). a+b=c and just like before:

and so:
and so:

etc like before:

etc, or
and then again we end to the same
IIb). a+b=N+c In a very analogous way, we get the same conclusion

We will prove now that N=6q+1.
Proof: From the following three relations, one at most can be true:

Indeed, if we really had, e.g.:


which is false
Therefore we assume, e.g.:

Now if N=6q-1, by sentence 1a we would have

which is false, because N doesn't divide either a-b or a+b as we have shown

Now we can show that
Indeed, the first two hold by assumption. If we also had


and since N doesn't divide b, we come to N=3, which is rejected by assumption.

Now we will prove the uniqueness of a triple (a,b,c), that is, given two members of the triple,say a & b,the third is uniquely defined
So if (a,b,c) is a triple and there was a second triple (a,b,c') then obviously:
(or vice versa)
which leads to:


Now we carry the relations from a,b,c to X,Y,Z.
sentence 3) N^2|Z^{3N}+X^{3N}" />
In the end of step 1 we proved:

and so:

But,as we showed:

In the same way we prove:

Step 3. The assumption that ? doesn't divide X,Y or Z must be abandoned
Let W=X-Y and so:
From relation (1):

and because (see relation (2)):

it follows that:
(because of (5))

and eliminating denominators:

???? N can't divide W, or (5) would yield N|3XY and since N doesn't divide 3 we conclude (sentence 5)
r\:N|Y" />
which is false. Thus, (6) gives, according to sentence 5 again:
Now's the time to use the only sentence that was unknown in the 17th century. It was proved by Mirimanoff in 1909:
In the first case of Fermat's last theorem,if:


Using this and (7), we immediately get:
But by Fermat's little theorem:

otherwise we would have

false, as we proved.
Therefore ( gives:
(??????? ??)
N^2Q+(a-b)^N-AN-a+BN+b ?? N^2| (a-b)^N-AN-a+BN+b
???? N^2|X^{N-1}-1 ?? N^2|X^N-X=(AN+a)^N-AN-a ??(??????? ??) N^2| a^N-AN-a ??? ????? N^2|b^N-b-BN ??? ???
N^2| (a-b)^N-a^N+b^N ??? ??????? ?? ??? ???????????? ??? a,b,c ??? ???????,??? b=c??N|c-b??N|a, ?????


????????????: 6
???????: ???? ??? 10, 2012 1:33 am
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