Interesting lemma

  • #1
Interesting lemma

If P.Ferma knew the proof of the following lemma
"Lemma. If numbers a, b, c, have no common factor, numbers c-a and c-b are also mutually-prime and n is odd, then the numbers (c^n-a^n)/(c-a) and (c^n-b^n)/(c-b) are also mutually-prime",
then with its aid it is possible to briefly and simply prove Fermat's last theorem.

Actually, in the Fermat’s equality (where numbers a, b, c have no common factor and n is odd) numbers c-a and c-b, obviously, mutually-prime. And then from the Fermat's little theorem it follows that with prime q>2c the numbers c^(q-1)-a^(q-1) and c^(q-1)-b^(q-1) are multiple by q.
And since, according to lemma, the numbers (c^(q-1)-a^(q-1))/(c-a) and (c^(q-1)-b^(q-1))/(c-b) are mutually-prime (i.e. have no common factor), then one of the numbers c-a and c-b is divided by q (>2c>c-b>c-a), i.e., the solution of the Fermat’s equation is not integer.

It remains to learn, who and when proved lemma.
 

Answers and Replies

  • #2
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
I hope nobody proved that lemma; it's not true!

Here's the smallest counterexample I could find:

a = 2
b = 2
c = 3
n = 3

a, b, c don't have a common factor.

c-a = 1
c-b = 1

These don't have a common factor

(c^n - a^n) / (c-a) = 19
(c^n - b^n) / (c-b) = 19

These do have a common factor.


Here's the smallest I could find with a and b unequal:

a = 1
b = 4
c = 9
n = 3

These don't have a common factor.

c-a = 8
c-b = 5

These don't have a common factor.

(c^3 - a^3) / (c-a) = 91
(c^3 - b^3) / (c-b) = 133

These do have a common factor: 7.


Here's another:

a = 5
b = 6
c = 17
n = 3

c-a = 12
c-b = 11

(c^3 - a^3) / (c-a) = 399
(c^3 - b^3) / (c-b) = 427

These do have a common factor: 7.
 
  • #3
I hope nobody proved that lemma; it's not true!

Here's the smallest I could find with a and b unequal:

a = 1
b = 4
c = 9
n = 3

These don't have a common factor.

c-a = 8
c-b = 5

These don't have a common factor.

(c^3 - a^3) / (c-a) = 91
(c^3 - b^3) / (c-b) = 133

These do have a common factor: 7.
Outstanding counterexamples!

Therefore I make correction to the Lemma:
the additional condition: a+b=c.
By the way, now to prove the lemma it is much easier.
 
  • #4
matt grime
Science Advisor
Homework Helper
9,395
3
Then feel free to post the proof.
 
  • #5
Lemma

Then feel free to post the proof.
Here are the minimum requirements for the lemma, necessary for the brief proof of the FLT: "Lemma. If integers a, b, c have only one common divisor 1, a+b=c and numbers (c^n-a^n) and (c^n-b^n) have common divisor d>2, then with n>1 one of the numbers c-b, c-a, a+b, c+b, c+a, a-b is divided by d ".
 
  • #6
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
Outstanding counterexamples!

Therefore I make correction to the Lemma:
the additional condition: a+b=c.
By the way, now to prove the lemma it is much easier.
No it's not.

a = 3, b = 10, c = 13, n = 3 is another counterexample.
a = 5, b = 12, c = 17, n = 3 is another.

If you're guessing, call it a conjecture. You can't call something a lemma until you have actually proven it.



Here are the minimum requirements for the lemma, necessary for the brief proof of the FLT: ...
matt asked you to post the proof. That's not a proof.
 
  • #7
No it's not.

a = 3, b = 10, c = 13, n = 3 is another counterexample.
a = 5, b = 12, c = 17, n = 3 is another.
Your counterexamples speak about the accuracy of the Lemma:
10-3=7,
12-5=7.
Consequently, it is possible to begin the search for its proof.
 
  • #8
1,074
1
Your counterexamples speak about the accuracy of the Lemma:
10-3=7,
12-5=7.
Consequently, it is possible to begin the search for its proof.
What do you mean by accuracy of the lemma? If there exists A counter example then it isn't true.
 
  • #9
What do you mean by accuracy of the lemma? If there exists A counter example then it isn't true.
Here is the Lemma, which easily proves and which is sufficient for the brief proof of the FLT:
Lemma. If integers a, b, c have only one common divisor 1, a+b=c, number (c^n-a^n) and (c^n-b^n) have common prime divisor d>3c^2, in base d numbers a^n, b^n, c^n finish by digit 1, then with n>1 either number (a^n-b^n)/(a+b) is not divided by d, or number 3c^2 is divided into d ".

to be continued
 
Last edited:
  • #10
Lemma And Its Proof

to be continued
Lemma.
If integers a, b, c have only one common divisor 1, a+b=c, and number (c^n-a^n) and (c^n-b^n) have common prime divisor d>3c^2, and in base d numbers a^n, b^n, c^n finish by digit 1, then with n>1 either number (a^n-b^n)/(a+b) is not divided by d, or number 3c^2 is divided by d.

Proof.
Let us assume that number (a^n-b^n)/(a+b) is divided by d.
Then number S=(c^n-b^n)/(c-b)+(c^n-a^n)/(c-a)+(a^n-b^n)/(a+b), or
(c^n-b^n)/a+(c^n-a^n)/b+(a^n-b^n)/c = [c^(n+2)+b^(n+2)+(a+b)c^(n+1]/abc=
=[c^2+b^2+a(c+b)]/abc, is divided by d.
But since number a^n, b^n, c^n finish by number 1, then the number s = c^2+b^2+a(c+b), where 0<s<d, is single-digit number and, therefore, number S is NOT DIVIDED by d.
 
  • #11
Elementary proof of Fermat's last theorem

Lemma.
If integers a, b, c have only one common divisor 1, a+b=c, and number (c^n-a^n) and (c^n-b^n) have common prime divisor d>3c^2, and in base d numbers a^n, b^n, c^n finish by digit 1, then with n>1 either number (a^n-b^n)/(a+b) is not divided by d, or number 3c^2 is divided by d.
Elementary proof of Fermat's last theorem

According to known theorem from the theory of integers, for any prime n there exist infinite great number of the prime numbers d of form d=pn+1, where p is integer. (This theorem is the simple consequence from the fact that, if numbers a and b are mutually-prime and n prime, then each divisor of the number (a^n+b^n)/(a+b) takes form pn+1).

Proof FLT:
Let us take simple d>c^(4n). Then, according to Fermat's little theorem, the numbers C^p-B^p, C^p-A^p, A^p-B^p, where C=c^n, B=b^n; A=a^n, are divided by d. And since A+B=C, then, according to Lemma, number 3C^2 is divided by d>3C^2. Consequently, the solution of Fermat’s equation is not integer.
 
Last edited:
  • #12
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
Can't you just stop posting this nonsense?
 
  • #13
Can't you just stop posting this nonsense?
OK!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
 
  • #14
Lemma is not true

Lemma.
If integers a, b, c have only one common divisor 1, a+b=c, and number (c^n-a^n) and (c^n-b^n) have common prime divisor d>3c^2, and in base d numbers a^n, b^n, c^n finish by digit 1, then with n>1 either number (a^n-b^n)/(a+b) is not divided by d, or number 3c^2 is divided by d.
Lemma is not true. New version of the lemma:
If in the simple base q the mutually-simple numbers A^p, B^p, C^p (where C=c^n, B=b^n, A=a^n, prime n>2 and p=/q-1) finish by digit 1, then either A+B and A-B or C+B and C-B, or C+A and C-A have common divisor not equal to 1.
Today I have not its proof.

Time out
 
  • #15
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
I repeat, if you don't have a proof, then the correct term is conjecture. You cannot (correctly) call it a lemma until it has been proven.
 
  • #16
Gib Z
Homework Helper
3,346
5
This guys tried to post the same thing in scienceforums.net, except there he blatantly said he was proving FLT. He was surprised when no one took him seriously >.<

Mate its good your trying, but wait till you learn abit more math. Fermat would have known this stuff.
 

Related Threads on Interesting lemma

  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
14
Views
3K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
5
Views
4K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
3
Views
710
  • Last Post
Replies
1
Views
715
  • Last Post
Replies
6
Views
3K
Top