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September 15th, 2017, 02:00 AM  #1 
Senior Member Joined: Oct 2011 Posts: 136 Thanks: 1  Fermat's theorem. Proof by 2 operations Fermat's theorem. Proof by 2 operations The essence of the contradiction. The hypothetical Fermat's equality is contradictory between the second digits of the factors of the number $А$. All calculations are done with numbers in base n, a prime number greater than 2. The notations that are used in the proofs: $A'$, $A''$ – the first, the second digit from the end of the number $A$; $A_2$ is the twodigit ending of the number A (i.e. $A_2=A$ mod $n^2$); Consider the Fermat’s equality in the base case (its properties 2°3° are proved here: The Proof of Fermat's Last Theorem for the Base Case, viXra.org ePrint archive, viXra:1707.0410) for coprimes positive $A$, $B$, $C$; $A'≠0$, prime $n$, $n>2$: 1°) $A^n=C^nB^n$ [$=(CB)P$], where (as is known) 2°) $A'≠0$, $CB=a^n$, $P=p^n$, $A=ap$, $p'=1$, $a'≠0$, $(a^n)'=a'$, $(a'^{n1})'=1$ (Fermat's small theorem); 3°) $(A+BC)_2=0$, from here 3a°) $(ap)_2=(a^n)_2$ and therefore 3b°) $p_2=(a^{n1})_2$. 4°) If $p''=0$, then we multiply term by term equality 1° by such $g^{nn}$, that $p''≠0$. Properties 2°3° are preserved, and we leave the notation of the numbers as before. And now the proof itself FLT. We represent the endings $a_2$ and $p_2$ in the form: $a2=(xn+a'^n)_2$ and $p_2=p''n+1$, where x and y are digits. First we substitute these values of the endings in the lefthand side of the equality 3a°: 5°) $[(xn+a'^n)(p''n+1)]_2=(a'^n)_2$, from here 5a°) $(a'np''n+xn)_2=0$, or (see 2°) $a'p''+x=0$ (mod n). Now, we substitute the value of $a_2$ in the righthand side of the equality 3b°: 6°) $(xn+a'n)^{n1}_2=[(n1)xna'^{n2}+1]2=(nxa'^{n2}+1)_2=(nxa'^{n1}/a'+1)_2$. From 3b° we have: 6a°) $xa'^{n1}/a'+p''=0$ (mod $n$), or $xa'^{n1}+a'p''=0$ (mod $n$), or $x+a'p''=0$ (mod $n$), It follows from 5a° and 6a° that $x=y=0$, which contradicts to 2° and 4°. From this contradiction follows the truth of the FLT. (Mezos, France. 4 September 2017) 
September 16th, 2017, 03:44 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,248 Thanks: 887 
What? That is not the way "proof by contradiction" works! "Proof by contradiction" starts by assuming that what is to be proved is not true then arrives at two statements that contradict one another. Since two contradictory statements cannot both be derived from a true statement, it follows that the original assumption, that what's to be proved is not true, must be false. And therefore that what was to be proved is true. Here, you don't do that. You assert 2° and 4° which you say have already been proved (I will grant you that) and then prove 5a° and 6a° which you say contradicts 2° and 4°. But you never assert that "Fermat's last theorem" is false to derive those. If you have done everything correctly, then you have disproved Fermat's last theorem! Oh, wait. I just realized this was posted by Victor Sorokin. What a waste of my time! Last edited by skipjack; February 27th, 2018 at 03:58 AM. 
September 17th, 2017, 01:30 AM  #3  
Senior Member Joined: Oct 2011 Posts: 136 Thanks: 1  Quote:
Last edited by skipjack; February 27th, 2018 at 04:00 AM.  
September 17th, 2017, 04:11 AM  #4 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,600 Thanks: 546 Math Focus: Yet to find out.  
September 17th, 2017, 04:28 AM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 2,011 Thanks: 1044  
September 23rd, 2017, 01:38 PM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,248 Thanks: 887 
Ah! It took me a moment to figure out what "method from the prive" meant since I did not know what "prive" meant. Now, thanks to romsek, I realize it is a "method from the privy"! All is clear now.

September 23rd, 2017, 01:47 PM  #7 
Senior Member Joined: Sep 2015 From: USA Posts: 2,011 Thanks: 1044 
we aim to please. ... especially in the privy 
February 26th, 2018, 07:51 PM  #8 
Senior Member Joined: Oct 2011 Posts: 136 Thanks: 1 
Dear sirs, R E M I N D that elementary proof of the FLT for a prime n>2 was found on December 1, 2017 and published on December 29, 2017 on the site: http://vixra.org/pdf/1712.0660v1.pdf Tools: Fermat's little theorem and the binomial theorem. The volume of the proof (without theory): 14 lines. In Russia, 20,500 schoolchildren and students read the proof (math.luga.ru : Форум  Главная страница). The author does not provide for any petitions either at the moment or in the future. The possible award will be given to the employee of the University who will make the official presentation of the proof (without the participation of the author, but consultations are guaranteed). Mezos (France) February 27, 2018 
February 26th, 2018, 09:32 PM  #9  
Senior Member Joined: Oct 2009 Posts: 430 Thanks: 144  Quote:
 
April 19th, 2018, 05:31 PM  #10 
Newbie Joined: Sep 2013 From: Reno,NV Posts: 14 Thanks: 1 Math Focus: Number Theory 
what does 2° mean? not the line but the "degree" symbol?


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