Iatco Sergiu. Simplification of prove of Fermat theorem
## Iatco Sergiu. Simplification of prove of Fermat theorem

R.Moldova

District Rascani

Village Recea

itsergiu@yahoo.com

Date: 4 May 1999

Dear Sir,

I am an amateur mathematician. First time I read about Fermat's last
theorem when I was 15 years old. Just like other people from the beginning I
dreamt to prove one day it. Last year I found out that A.Wiles and R.Taylor
proved it. I read this proof and I found it (just like other people) too
complex. I analysed the Fermat's last theorem and I succeed to simplify it
as follows:

Let have Fermat's equation:

a^{n}+b^{n}=c^{n} , where n>2 (1)

Because c=p1*...*pt, where pi - prime number, equation (1) becomes:

a^{n}+b^{n}= p1^{n}*...*pt^{n}
(2)

If exist such pi for which a1^{n}+b1^{n}=
pi^{n} (3) has solutions then these solutions are also solutions
for (2)

Let r= p1*...*pi-1*pi+1*...*pt

Multiplying (3) with r^{n }we have:

(r*a1)^{n}+(r*b1)^{n}= pi^{n}, let a=r*a1
b=r*b1

a^{n}+b^{n}= p1^{n}*...*pt^{n}
- what had to be proved

What must be proved but I could not is that (2) has solutions only if
(3) has solutions

**Theorem 1 (unproved by me)
**

a^{n}+b^{n}= p1^{n}*...*pt^{n}
- has sloutions only if a1^{n}+b1^{n}=pi^{n
}Let return to Fermat's equation (1) :

a^{n}+b^{n}=c^{n
}If (1) is divided by c^{n} it becomes:

(*a)^{n+}(*b)^{n}=1

can be definited as:

a) =d/10^{k}, where ^{ }d,k N

b) =t/10^{k}*(10^{m}-1), where ^{ }t,m,k
N

Therefore (1) becomes

(a*d)^{n}+(b*d)^{n}=(10^{k})^{n
} (4)

(a*t)^{n}+(b*t)^{n}=10^{mn}*(10^{k}-1)^{n}
(5)

or,

a^{n}+b^{n}=10^{kn } (6)

a^{n}+b^{n}=10^{mn}*(10^{k}-1)^{n}
(7)

Therefore in order to prove (1) must be proved that (6) and (7) do not
have solutions for n>2.

Let solve first a^{n}+b^{n}=10^{kn
}Accordingly with **theorem 1** (6) has solution only if

a^{n}+b^{n}=5^{n} or
a^{n}+b^{n}=2^{n}

a^{n}+b^{n}=2^{n} - does not has
solutions for n>2

a^{n}+b^{n}=5^{n} - does not has
solutions for n>2

Let now solve
a^{n}+b^{n}=10^{mn}*(10^{k}-1)^{n},
where n>2 a,b,k,m,nN

Accordingly with **theorem 1** (7) has solutions only if

a^{n}+b^{n}=10^{mn} or
a^{n}+b^{n}=(10^{k}-1)^{n
}a^{n}+b^{n}=10^{mn} has already
been examined

Therefore must be proved:

a^{n}+b^{n}=(10^{k}-1)^{n} (8)

Regretfully I could not prove (8).

Finally in order to prove Fermat's theorem must be proved **theorem 1** and
**equation (8)**.

I will be happy if you publish my work and after that somebody will
come with a simply proof like Fermat's ones.

Of course you should publish it only if I am not wrong.

I will be grateful if you give me an answer to my letter.

Thank you,

Respectfully,

*Sergiu Iaþco*

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