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August 28th, 2015, 09:06 AM   #51
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From: Barto PA

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I just quoted the relevant parts of your text.
Anyone interested in this thread can easily go
back to see exactly what was written.

You wrote: "The v^0 = w^0 = 1 does not mean
v = w = 1." We are evidently not using the
same rules of arithmetic and logic - and that
seems to be the basic problem.

I wrote: "It is certainly possible that at least
one of a, b in a^n + b^n = c^n could be prime
if n > 2."
You replied: "It's absolutely unfounded
statement." ... "If you can deduce your
'certainly possible' this way I would accept it."
Consider this: How exactly would you _prove_
neither a nor b could be prime _without_
supposing FLT is true? I merely pointed out
what is possible without that supposition.
The burden of proof is not mine.

You wrote: "But otherwise v and w (not v^n
and w^n) can easily be primes but > 1." So
now that's clear.

I don't think it's necessary to go further. You
obviously disagree with my and al-mahed's
analyses of your argument. Why not take his
suggestion to post on another forum? If you
do, however, I believe you will get similar
objections. Or, request Mr. Greathouse's
opinion - he may be willing to comment.
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August 28th, 2015, 10:36 AM   #52
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Quote:
Originally Posted by uvkajed View Post
I just quoted the relevant parts of your text.
Anyone interested in this thread can easily go
back to see exactly what was written.

You wrote: "The v^0 = w^0 = 1 does not mean
v = w = 1." We are evidently not using the
same rules of arithmetic and logic - and that
seems to be the basic problem.
What "rules of arithmetic and logic" are you using? The ones I am used to say that any number to the 0 power is equal to 1.

Quote:
I wrote: "It is certainly possible that at least
one of a, b in a^n + b^n = c^n could be prime
if n > 2."
You replied: "It's absolutely unfounded
statement." ... "If you can deduce your
'certainly possible' this way I would accept it."
Consider this: How exactly would you _prove_
neither a nor b could be prime _without_
supposing FLT is true? I merely pointed out
what is possible without that supposition.
The burden of proof is not mine.

You wrote: "But otherwise v and w (not v^n
and w^n) can easily be primes but > 1." So
now that's clear.

I don't think it's necessary to go further. You
obviously disagree with my and al-mahed's
analyses of your argument. Why not take his
suggestion to post on another forum? If you
do, however, I believe you will get similar
objections. Or, request Mr. Greathouse's
opinion - he may be willing to comment.
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August 28th, 2015, 11:06 AM   #53
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Quote:
Originally Posted by Country Boy View Post
What "rules of arithmetic and logic" are you using? The ones I am used to say that any number to the 0 power is equal to 1.
I think you're talking past each other here. uvkajed's claim was that, even though v^0 = 1 and w^0 = 1 this doesn't mean that v = w.
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August 28th, 2015, 12:56 PM   #54
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Quote:
Originally Posted by CRGreathouse View Post
I think you're talking past each other here. uvkajed's claim was that, even though v^0 = 1 and w^0 = 1 this doesn't mean that v = w.
Quote:
Originally Posted by uvkajed View Post
After checking again, I have to retract my
statement that the argument holds if n = 1.
That is because both v and w must equal 1 by
equation 10, but then f = v = 1 and k = w = 1
contradicts Lemma 3.
Quote:
Originally Posted by uvkajed View Post
I just quoted the relevant parts of your text.
Anyone interested in this thread can easily go
back to see exactly what was written.

You wrote: "The v^0 = w^0 = 1 does not mean
v = w = 1." We are evidently not using the
same rules of arithmetic and logic - and that
seems to be the basic problem.

I wrote: "It is certainly possible that at least
one of a, b in a^n + b^n = c^n could be prime
if n > 2."
You replied: "It's absolutely unfounded
statement." ... "If you can deduce your
'certainly possible' this way I would accept it."
Consider this: How exactly would you _prove_
neither a nor b could be prime _without_
supposing FLT is true? I merely pointed out
what is possible without that supposition.
The burden of proof is not mine.
It looks like all this discussion is based on misunderstanding. Because the proof is started with assumption that a^n+b^n=c^n is true. Then the polynomial expressions for a, b, c required to satisfy this equation are deduced. Their correctness is proved for n=2 (Eq.26). Then it's proved that polynomials don't satisfy equation when n>2.
This was opposed by pledge that there are other numbers that don't satisfy the equation. Certainly they are. FLT is true for all integers but only these corresponding to deduced polynomials should be tried.
It's what I meant opposing suggestions about possible other numbers.
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September 1st, 2015, 08:50 AM   #55
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What I am trying to understand is: Which
part of my reply to Mr. Pogorsky did Country
Boy not understand? Of course if w^0 = v^0
= 1 it then follows w = v = 1. Mr. Pogorsky
seems to believe it does _not_ follow (see post
#50). The only real issue here seems to be
reading and comprehending what was written.
Also, it follows from equation (10) if u = 0
when n = 1 then p = q = 1 is necessary; then
from (7a) and (7b), a = vp and b = wq with all
of p, q, v, w = 1, it follows that a = b = 1. The
equations should of course reduce to a + b = c
if n = 1 and they do not - that is what I missed
when I first checked the algebra.

Again, Mr. Pogorsky said my statement that
at least one of a, b can be prime if n > 2 is
"absolutely unfounded". No - it is certainly
not at all unfounded because it _is_ possible
unless he can _prove_ it is not. He also stated
that v and w can be prime, so I do not see any
problem with my assertion that if a is prime
then v must equal a if v divides a.
[This is what I meant when I had stated in a
previous post that not all possibilities had
been considered: One can not just _assume_
a^n and/or b^n have two _unique_ factors,
that is, a and b are not necessarily composite.
This all comes back to the improper statement,
proof, and application of Lemma-3. I said
near the beginning of this thread that the
only thing this lemma seems to prove is 1 = 1,
and that is hardly a basis for attempting to
prove FLT.]

Perhaps the mathematician whom he claimed
reviewed his paper can clarify any objections
that have been given on this forum - clearly,
no one here has succeeded doing that to Mr.
Pogorsky's satisfaction.

I might soon be posting my own totally revised
(and completely different than my last alleged)
proof of FLT (everyone should have a backup
proof of FLT just in case!); so I must refrain
from any further comment on Mr. Pogorsky's
paper. If he still believes his argument is a
proof he should submit it to a professional
mathematics journal.
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September 1st, 2015, 09:19 PM   #56
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Quote:
Originally Posted by uvkajed View Post
Of course if w^0 = v^0
= 1 it then follows w = v = 1. Mr. Pogorsky
seems to believe it does _not_ follow (see post
#50). The only real issue here seems to be
reading and comprehending what was written.
Also, it follows from equation (10) if u = 0
when n = 1 then p = q = 1 is necessary; then
from (7a) and (7b), a = vp and b = wq with all
of p, q, v, w = 1, it follows that a = b = 1. The
equations should of course reduce to a + b = c
if n = 1 and they do not - that is what I missed
when I first checked the algebra.
I was trying to understand how you come to conclusion that v=w=1 when n=1. Even CRGreathouse thought that you don't mean this but you continue insist on it referring not once to equation (10). So the only way to find truth is to explore Eq.(10). When n=1 it becomes
(p-v^0)/w=(q-w^0)/v=u;
(p-1)/w=(q-1)/v=u;
v(p-1)=w(q-1)=uwv
Did you concluded from this v=w=1?
Now we bring here u=0 that follows from uwv+v+uwv+w=uwv+v+w.
Since neither v nor w equal to zero the only conclusion - hurrah! -p=q=1.
But when n=1 we have p=uw+1 and q=uv+1 and we have pretty zeros in parentheses
None of the above steps impose any restriction on value of v and w. So I don't know which of them your conclusions are based on. Maybe you would explain, maybe not. Because

Quote:
Again, Mr. Pogorsky said my statement that
at least one of a, b can be prime if n > 2 is
"absolutely unfounded". No - it is certainly
not at all unfounded because it _is_ possible
unless he can _prove_ it is not.
It looks like justice in some countries where burden of proof (always unsuccessful) actually is on defendant though charges may be completely unfounded.
The presented polynomial expressions for a, b, c were obtained through succession of strict transformations (no errors still disclosed). Now somebody (in general) declares that he/she believes that there are kinds of a, b, different from deduced and demands from me to prove that the belief is wrong. By the way what does it mean at all? FLT is true for all numbers. My polynomials would satisfy the a^n+b^n=c^n if the theorem were not true. Is it asserted that prime a or b would satisfy equation too?


Quote:
He also stated that v and w can be prime, so I do not see any
problem with my assertion that if a is prime
then v must equal a if v divides a.
[This is what I meant when I had stated in a
previous post that not all possibilities had
been considered: One can not just _assume_
a^n and/or b^n have two _unique_ factors,
that is, a and b are not necessarily composite.
There's nothing assumed except a^n+b^n=c^n is true, then everything is deduced. To the contrary all your assertions are taken from air and you require me this airborne staff to disprove.

Quote:
This all comes back to the improper statement,
proof, and application of Lemma-3. I said
near the beginning of this thread that the
only thing this lemma seems to prove is 1 = 1,
and that is hardly a basis for attempting to
prove FLT.]
The essence of Lemma-3 is simple and well known maybe for centuries. All this fuss around it is worthless and covers unwillingness or inability to find real reasons for so much wanted disproof. None of your assertions can be called a reason and when you tried to give reasons (case n=1) they appeared completely wrong.

Last edited by McPogor; September 1st, 2015 at 09:46 PM.
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September 4th, 2015, 01:29 AM   #57
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Quote:
Originally Posted by McPogor View Post
I was trying to understand how you come to conclusion that v=w=1 when n=1. Even CRGreathouse thought that you don't mean this but you continue insist on it referring not once to equation (10). So the only way to find truth is to explore Eq.(10). When n=1 it becomes
(p-v^0)/w=(q-w^0)/v=u;
(p-1)/w=(q-1)/v=u;
v(p-1)=w(q-1)=uwv
Did you concluded from this v=w=1?
Now we bring here u=0 that follows from uwv+v+uwv+w=uwv+v+w.
Since neither v nor w equal to zero the only conclusion - hurrah! -p=q=1.
But when n=1 we have p=uw+1 and q=uv+1 and we have pretty zeros in parentheses
None of the above steps impose any restriction on value of v and w. So I don't know which of them your conclusions are based on. Maybe you would explain, maybe not. Because


It looks like justice in some countries where burden of proof (always unsuccessful) actually is on defendant though charges may be completely unfounded.
The presented polynomial expressions for a, b, c were obtained through succession of strict transformations (no errors still disclosed). Now somebody (in general) declares that he/she believes that there are kinds of a, b, different from deduced and demands from me to prove that the belief is wrong. By the way what does it mean at all? FLT is true for all numbers. My polynomials would satisfy the a^n+b^n=c^n if the theorem were not true. Is it asserted that prime a or b would satisfy equation too?




There's nothing assumed except a^n+b^n=c^n is true, then everything is deduced. To the contrary all your assertions are taken from air and you require me this airborne staff to disprove.



The essence of Lemma-3 is simple and well known maybe for centuries. All this fuss around it is worthless and covers unwillingness or inability to find real reasons for so much wanted disproof. None of your assertions can be called a reason and when you tried to give reasons (case n=1) they appeared completely wrong.
If you actually read my last post you would
have noticed I _did_ refer to equation (10).
Nevertheless, one more time - just to clarify:

When n = 1 then w = v = 1 by equation (10). Then p = q = 1 is _required_ by (10) if u is to be
zero. Now we have all of w, v, p, q = 1. Then
by (7a) and (7b), a = vp = 1 and b = wq = 1.
You can _not_ change the values of w, v, p, q
once they have been determined by (10) for
a particular case, and apparently that is what
you have done because now you write: "But when n = 1 we have p = uw + 1 and q = uv + 1
and we have pretty zeros in parentheses
None of the above steps impose any
restriction on value of v and w." Exactly how
are those statements to be interpreted?

As for equations (4a) and (4b), you seem to
believe that the right hand sides of each are
_always_ the product of two _distinct_ factors,
and that is not true if either a or b is prime.
Again, it is not _my_ responsibility to prove neither a nor b can be prime. That at least
one of a, b could be prime is certainly true if
n = 1 or n = 2. To _assume_ it is _not_ true if
n > 2 is totally unwarranted, and you either
assumed it or you did not even consider the
possibility - it really doesn't matter which.
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September 4th, 2015, 09:22 AM   #58
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Quote:
Originally Posted by uvkajed View Post
If you actually read my last post you would
have noticed I _did_ refer to equation (10).
It has been acknowledged. I'm not sure you tried to understand what I wrote especially because you continue to declare after all

Quote:
When n = 1 then w = v = 1 by equation (10). Then p = q = 1 is _required_ by (10) if u is to be
zero. Now we have all of w, v, p, q = 1. Then
by (7a) and (7b), a = vp = 1 and b = wq = 1.
I transformed this unfortunate (but true) Eq.(10) step by step trying to find where this conclusion w=v=1 may come from. There's not any clue. If you see one show at what step it is concluded. Or am I required to prove it's false according to your distorted logic?
Quote:
You can _not_ change the values of w, v, p, q
once they have been determined by (10) for
a particular case, and apparently that is what
you have done because now you write: "But when n = 1 we have p = uw + 1 and q = uv + 1
and we have pretty zeros in parentheses
None of the above steps impose any
restriction on value of v and w." Exactly how
are those statements to be interpreted?
They need to be not interpreted but understood. Since a=uwv+v^n=pv it becomes p=uw+1 etc. when n=1. And nothing except your unfounded statements proves that v and w cannot be of any value.

Quote:
As for equations (4a) and (4b), you seem to
believe that the right hand sides of each are
_always_ the product of two _distinct_ factors,
and that is not true if either a or b is prime.
Again, it is not _my_ responsibility to prove neither a nor b can be prime.
Again it is according to your distorted logic (reminding justice of totalitarian regimes). I have only to repeat that there's everything is deduced not believed.
Quote:
That at least
one of a, b could be prime is certainly true if
n = 1 or n = 2. To _assume_ it is _not_ true if
n > 2 is totally unwarranted, and you either
assumed it or you did not even consider the
possibility - it really doesn't matter which.
I would not stop on it but I'm curious again what does it mean: a or b -prime?
Would FLT be untrue in this case?
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September 6th, 2015, 04:42 AM   #59
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"I transformed this unfortunate (but true) Eq.(10) step by step trying to find where this conclusion w=v=1 may come from. There's not any clue. If you see one show at what step it is concluded. Or am I required to prove it's false according to your distorted logic?" (...) "And
nothing except your unfounded statements
proves that v and w cannot be of any value."

SERIOUSLY??? You can't make sense of a
set of equations YOU derived??? Incredible.

How many times is it necessary for something
to be explained before you finally understand
it? Or is that even possible?

AGAIN!: In equation (10) you have the terms
w^(n-1) and v^(n-1) in the numerators of the
fractions. The last time I bothered checking,
when n = 1 then n - 1 = 1 - 1 = 0. And then
(again!) w^0 = v^0 = 1. And that is _exactly_
where w = v = 1 comes from! Then (p - 1)/1 =
(q - 1)/1 = u, and if u = 0 when n = 1 then
p = q = 1 is _necessary_! And don't forget
YOU are the one who said in post #39 that
u = 0 when n = 1. Now I imagine the _next_
thing you will try telling me is w^0 and v^0
does NOT mean w = 1 and v = 1. Don't even
go there.

"I would not stop on it but I'm curious
again what does it mean: a or b -prime?"
Suggestion: Check the definition of 'prime
number'. Hint: Those numbers have no
factors.

"Would FLT be untrue in this case?"
You figure it out. After all, you're the one
who 'proved' it.

So: My logic is "distorted" and my statements
are "unfounded". Hmm... if anyone's logic
is distorted it is yours, but you are correct
in one respect: You are definitely clueless.
Enough is enough already.
End of discussion.
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September 6th, 2015, 10:13 AM   #60
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Quote:
Originally Posted by uvkajed View Post
"I transformed this unfortunate (but true) Eq.(10) step by step trying to find where this conclusion w=v=1 may come from. There's not any clue. If you see one show at what step it is concluded. Or am I required to prove it's false according to your distorted logic?" (...) "And
nothing except your unfounded statements
proves that v and w cannot be of any value."

SERIOUSLY??? You can't make sense of a
set of equations YOU derived??? Incredible.

How many times is it necessary for something
to be explained before you finally understand
it? Or is that even possible?

AGAIN!: In equation (10) you have the terms
w^(n-1) and v^(n-1) in the numerators of the
fractions. The last time I bothered checking,
when n = 1 then n - 1 = 1 - 1 = 0. And then
(again!) w^0 = v^0 = 1. And that is _exactly_
where w = v = 1 comes from! Then (p - 1)/1 =
(q - 1)/1 = u, and if u = 0 when n = 1 then
p = q = 1 is _necessary_! And don't forget
YOU are the one who said in post #39 that
u = 0 when n = 1. Now I imagine the _next_
thing you will try telling me is w^0 and v^0
does NOT mean w = 1 and v = 1. Don't even
go there.

"I would not stop on it but I'm curious
again what does it mean: a or b -prime?"
Suggestion: Check the definition of 'prime
number'. Hint: Those numbers have no
factors.

"Would FLT be untrue in this case?"
You figure it out. After all, you're the one
who 'proved' it.

So: My logic is "distorted" and my statements
are "unfounded". Hmm... if anyone's logic
is distorted it is yours, but you are correct
in one respect: You are definitely clueless.
Enough is enough already.
End of discussion.
Our wishes coincide this time. No more comments.
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