An Elementary Proof Of Both The Beal Conjecture And Fermat's Last Theorem.

[Don Blazys]

To: All who are following this thread.

 

This proof is so simple!

 

First...Factor the original equation.

 

Then...Substitute a "Blazys term" for the term containing [math]c[/math]

in both the original equation, and the factored equation.

 

Then... simply note that in order to allow [math]T=c[/math] (which must be allowable),

the expressions involving logarithms must first be "cancelled out",

and that they "cancel out" if and only if [math]z=1[/math] in the original equation,

and [math]z=2[/math] in the factored equation.

 

That's all there is to it! It's breathtaking!

 

However, Craig D felt that it needed to be challenged!

 

First he said that the expressions involving logarithms don't "cancel out" at [math]z=1[/math] and [math]z=2[/math].

 

In my humble opinion, he was wrong.

 

Then, he said that factoring the original equation results in "circular reasoning".

 

In my humble opinion, he was wrong again.

 

Now, does any Hypographer other than myself

understand my proof and what has occured here?

 

I'm frustrated because Craig D has not admitted that he is wrong,

and I don't know if I am wasting my time debating him here because

I don't know if anyone else here at Hypography actually understands the math.

 

I do know that some people are following this thread as if it were a "soap opera",

but who else here actually understands the math?

 

Please, I need some feedback on this!

I'm feeling very alone, and seriously considering moving on to another forum.

Also, if you hate me for any reason whatsoever, just say so, and I will go.

I don't want to be where I'm neither wanted nor welcome!

 

(If you don't want to go "on record" then send me an "E-mail" or "private message".)

 

Don.

[CraigD]

[math]c^z-b^y=a^x[/math], __________________________________________________(1)

 

Factoring (1) results in:

 

[math]\left(c^\frac{z}{2}+b^\frac{y}{2}\right)\left(c^\frac{z}{2}-b^\frac{y}{2}\right)=a^x[/math], ____________________________________(2)

I see that you have changed you proof, Don, to justify choosing “2” for the transformation of

[math]a^x +b^y =c^z[/math]

to

[math]a^{\frac{x}{2}} +b^{\frac{y}{2}} = c^{\frac{z}{2}}[/math]

 

:thumbs_up I applaud you, as this eliminates choosing “3” rather than “2”, the example I gave in post #3. It’s also, like many of your observations, though provoking.

 

:thumbs_do However, the choice of “2” remains unsupported by a theorem other than the one being proven, because if can be replaced with any positive integer power of 2, such as “4”. This leads to proving that

[math]a^4 +b^4 = c^4 [/math]

has integer positive solutions [math]a[/math], [math]b[/math], [math]c[/math], which contradicts Fermat’s last theorem (which states that [math]a^n +b^n =c^n[/math] has positive integer solutions [math]a[/math], [math]b[/math], [math]c[/math], [math]n[/math] only for [math]n<3[/math]). My previous question, intended to lead you to rethink you claims,

1) If there was known to exist a triple of integers [math]a,b,c > 1[/math] such that [math]a^3+b^3=c^3[/math], would the choice of [math]3[/math] in place of (or in addition to) [math]2[/math] in (2) be permitted?

 

2) How do you know there does not exist such a triple?

Can be rephrased, replacing “3” with “4” (or any positive integer power of 2):

1) If there was known to exist a triple of integers [math]a,b,c > 1[/math] such that [math]a^4+b^4=c^4[/math], would the choice of [math]4[/math] in place of (or in addition to) [math]2[/math] in (2) be permitted?

 

2) How do you know there does not exist such a triple?

Thus it’s still necessary, in your proof, to assume that Fermat’s last theorem has already been proven.

 

On a non-mathematical note, I appreciate that you have edited some of your previous posts to be less “silly” and ad hominem. However, I believe you’re still making unsupported and strange claims. In particular, claims such as

However, in over ten years, no one,

not even the math department at Princeton University,

has been able to find a "fatal flaw" in it!

Imply that some member the the faculty of the Math department of Princeton University has reviewed your proof, and stated that he or she finds it sound. If you intend to make this claim, you must back it up with a link or reference to such a statement. If you back up your claims, and avoid making strange, exaggerated claims (eg: one’s containing phrases like “absolutely irrefutable”, “incredible”, and “even a child can see”), I believe you’ll be taken more seriously and afforded more respect.

[Don Blazys]

To: Craig D,

 

Quoting Craig D:

On a non-mathematical note, I appreciate that you have edited some of your

previous posts to be less “silly” and ad hominem.

However, I believe you’re still making unsupported and strange claims.

In particular, claims such as

 

Quote:

Originally Posted by Don Blazys

However, in over ten years, no one,

not even the math department at Princeton University,

has been able to find a "fatal flaw" in it!

 

Imply that some member the the faculty of the

Math department of Princeton University has reviewed your proof,

and stated that he or she finds it sound.

If you intend to make this claim,

you must back it up with a link or reference to such a statement.

If you back up your claims, and avoid making strange,

exaggerated claims

(eg: one’s containing phrases like “absolutely irrefutable”,

“incredible”, and “even a child can see”),

I believe you’ll be taken more seriously and afforded more respect.

 

Respect is okay, but I don't really need it.

I prefer to be treated as just an ordinary man.

Besides, I'm older than most people I know,

so if someone doesn't respect their elders,

then that's their shortcoming, not mine.

It would be nice, however,

for my discovery to be taken a bit more seriously.

 

Anyway, phrases such as "absolutely irrefutable", "incredible", etc.

were just a misguided attempt at "promoting" an idea that will

ultimately succeed or fail on it's merits alone. I'll edit them out too.

(My wife is certain that I'm a "borderline autistic" with no "people skills" whatsoever,

and after all I've been through for the past ten years because of this proof,

I'm only now beginning to face the possibility that she may be right!)

 

As for Princeton University being aware of my proof, well, they are.

You see, last year, I made an "electronic submission"

(of a version of my proof similar to the one on my website)

to the Journal of the American Mathematical Society.

It wasn't in LaTex, so I rather expected that it would be

declined publication for that reason alone.

But you see, what I was really after was a "referees report",

so that I could make any necessary changes

before having to pay someone to prepare it in LaTex for me.

(I'm the sole support of my family and can't afford to waste any money.)

Anyway, after about four months, I recieved an e-mail from a professor at Princeton

telling me that my proof was declined publication.

I e-mailed him back, and explained that at this time, publication is not important,

and that what I really need is the "referees report".

He e-mailed me back telling me that "there is no referees report",

and that the recommendation to decline publication was "verbal".

Well, needless to say, I was absolutely livid!

The proof is only one page long,

and I waited four months for what amounted to nothing?!

I then kept on insisting and demanding that someone at Princeton read my proof,

and tell me if they can or cannot find a fatal flaw.

The last e-mail I recieved from them came from a rather high ranking editor.

It said:

 

Quoting final e-mail from Princeton University:

We are not in a position to check for correctness proposed solutions to famous problems,

nor are we able to provide professional help to amateurs in developing their ideas.

This is not to say that amateurs cannot make important contributions,

but the editorial board of a major journal is not equipped to provide help in this direction.

I would also point out that we decline to publish many submissions

whose correctness is not in question.

 

Now, if you only knew what a complete and utter nuisance I made of myself,

then you would know that had they found a fatal flaw,

then they would have been more than happy to point it out to me!

 

The only other time that I submitted my proof to a major journal was to the

Journal of the London Mathematical Society (University of Leeds) about ten years ago.

At that time, it was a handwritten manuscript entitled "On The General Case Of FLT".

The letter that I recieved from them said:

 

Quoting letter from the University of Leeds:

Although the referee gave your work some support,

journal space is very limited at this time,

and we must decline in favour of more highly recommended contributions.

 

Note that in neither case, did a referee find a "fatal flaw".

Thus, my proof remains an enigma.

 

I still have all those e-mails, and that letter, and if you would like copies,

I would be more than happy to mail them to you.

They constitute evidence that would be admissable in a court of law,

so I am definitely prepared to "back up" my claims.

Just tell me where you want them sent, and you will have them, I promise!

 

You know, in the past ten years, I have spent hundreds of hours discussing my proof

with many professors of mathematics, none of whom ever gave me a reason to quit!

Indeed, it was a professor with two degrees (physics and math) who suggested that

I put my ideas out on the internet.

 

If my idea is a little ahead of it's time and I don't get to see it succeed,

then my son and grand daughter will continue to "push it along" until it does.

 

Anyway, I will edit out all those parts also, but only because I trust your judgement on this,

and not because I made any claims that I know to be false.

 

Quoting Craig D:

However, the choice of “2” remains unsupported by a theorem other than the one being proven, because it can be replaced with any positive integer power of 2, such as “4”. This leads to proving that:

 

[math]a^4+b^4=c^4[/math]

 

has integer positive solutions [math]a,b,c[/math] which contradicts Fermat’s last theorem (which states that [math]a^n+b^n=c^n[/math] has positive integer solutions [math]a, b, c, n[/math] only for math [math]n<3[/math] ). My previous question, intended to lead you to rethink you claims,

 

Quote:

Originally Posted by CraigD

1) If there was known to exist a triple of integers [math]a, b, c, >1[/math] such that , [math]a^3+b^3=c^3[/math], would the choice of [math]3[/math] in place of (or in addition to) [math]2[/math] in (2) be permitted?

 

2) How do you know there does not exist such a triple?

 

Can be rephrased, replacing “3” with “4” (or any positive integer power of 2):

 

1) If there was known to exist a triple of integers [math]a,b,c>1[/math] such that [math]a^4+b^4=c^4[/math], would the choice of [math]4[/math] in place of (or in addition to) [math]2[/math] in (2) be permitted?

 

2) How do you know there does not exist such a triple?

 

Thus it’s still necessary, in your proof, to assume that Fermat’s last theorem has already been proven.

 

With all due respect to a fine mathematician, that's not true.

 

Any 4th power must be representable as a 4th power (where [math] z=4[/math]),

and the terms involving logarithms clearly show that

any such representation must automatically result in division by zero

and must therefore be impossible.

Thus, the proof, in it's present form, works perfectly, and needs no revision.

 

I should also add that proving the "general case" of a theorem

automatically proves all "special cases" of that same theorem,

but proving a "special case" of a theorem, (as did Wiles),

does not necessarily prove the "general case" (as did I).

Thus, my proof does not need to rely on Wile's proof

in any way, shape or form whatsoever.

 

It's really not that complicated!

All I did was prove that any value of [math]z>2[/math] would result in division by zero,

and that alone is sufficient to demonstrate that both

the Beal Conjecture and Fermat's Last Theorem are true.

 

Be that as it may, what you are demanding then,

is that the argument be made using equations

that do not involve any numbers whatsoever.

In other words, you are insisting on equations that are comprised of "variables only".

Well, for positive integer variables, the terms involving logarithms

are every bit as adaptable as ordinary algebraic terms,

(only they contain more "information" and actually require

that unity not be defined in terms of itself ! ).

Thus, it is possible to re-write the proof using the variable [math]q[/math],

instead of the number [math]2[/math], just as I mentioned at the end of post #8.

 

Here then, is the "Craig D Hypography version" of my proof,

that can't possibly involve "circular reasoning" because

the requirement that we allow [math]T=c[/math],

occurs simultaneously with

the requirement that we allow Pythagorean triples.

[Don Blazys]

An Elementary Proof Of Both The Beal Conjecture And Fermat's Last Theorem.

_______________________Craig D Hypography Version________________________

____________________________By: Don Blazys______________________________

 

The Beal Conjecture can be stated as follows:

 

For positive integer variables, if the terms comprising the equations:

 

[math]\left(\left(\frac{T}{T}\right)a^\frac{x}{q}\right)^q+\left(\left(\frac{T}{T}\right)b^\frac{y}{q}\right)^q=\left(\left(\frac{T}{T}\right)c^\frac{z}{q}\right)^q=\left(T\left(\frac{c}{T}\right)^{\left(\frac{\frac{\frac{z}{q}\ln©}{\ln(T)}-1}{\frac{\ln©}{\ln(T)}-1}\right)}\right)^q[/math] _______(1)

 

and:

 

[math]\left(\frac{T}{T}\right)a^x+\left(\frac{T}{T}\right)b^y=\left(\frac{T}{T}\right)c^z=T\left(\frac{c}{T}\right)^{\left(\frac{\frac{z\ln©}{\ln(T)}-1}{\frac{\ln©}{\ln(T)}-1}\right)}[/math] ___________________________(2)

 

are co-prime, then [math]x, y[/math] and [math]z[/math] can not all be greater than [math]2[/math].

 

______________________________Proof______________________________

 

Both [math]T=c[/math] and the formation of "Pythagorean triples" at [math]x=y=z=2[/math]

must be allowable, and can occur if and only if [math]z=q=2[/math] in (1) and [math]z=1[/math] in (2),

which results in both:

 

[math]\left(\left(\frac{T}{T}\right)a^\frac{x}{2}\right)^2+\left(\left(\frac{T}{T}\right)b^\frac{y}{2}\right)^2=\left(\left(\frac{T}{T}\right)c\right)^2=\left(T\left(\frac{c}{T}\right)\right)^2[/math] ____________________(3)

 

and:

 

[math]\left(\frac{T}{T}\right)a^x+\left(\frac{T}{T}\right)b^y=\left(\frac{T}{T}\right)c=T\left(\frac{c}{T}\right)[/math]. _____________________________________(4)

 

This proves both the Beal Conjecture, and Fermat's Last Theorem,

(since the latter is but the special case where [math]x=y=z[/math]).

[Don Blazys]

To: Craig D,

 

Actually, this is even more solid than before, and a nice simplification to boot!

 

Anyone can understand it now!

 

Even if you still think it's "somehow flawed", I like it !

 

Thanks for the idea.

 

By the way, I edited out all the "unsuitable" stuff that I could find

without losing any continuity whatsoever.

Please read the entire thread from beginning to end

and let me know if there is anything that I missed.

 

Don.

[Don Blazys]

The way I see it, a "formal proof" of the Beal Conjecture or Fermat's Last Theorem

isn't even necessary, because the moment we factor the equation:

 

[math]c^z-b^y=a^x[/math],

 

we get:

 

[math]\left(c^\frac{z}{2}+b^\frac{y}{2}\right)\left(c^\frac{z}{2}-b^\frac{y}{2}\right)=a^x[/math],

 

and since the above two equations can also be viewed as:

 

[math]T\left(\frac{c}{T}\right)^{\left(\frac{\frac{z\ln©}{\ln(T)}-1}{\frac{\ln©}{\ln(T)}-1}\right)}-\left(\frac{T}{T}\right)b^y=\left(\frac{T}{T}\right)a^x[/math]

 

and:

 

[math]\left(T\left(\frac{c}{T}\right)^{\left(\frac{\frac{\frac{z}{2}\ln©}{\ln(T)}-1}{\frac{\ln©}{\ln(T)}-1}\right)}+\left(\frac{T}{T}\right)b^\frac{y}{2}\right)\left(\left(\frac{T}{T}\right)c^\frac{z}{2}-\left(\frac{T}{T}\right)b^\frac{y}{2}\right)=\left(\frac{T}{T}\right)a^x[/math],

 

where [math]a, b[/math] and [math]c[/math] are "co-prime",

it is obvious that [math]T=c[/math] must be allowable,

and that this, in turn, requires that we first "cancel out"

the expressions involving logarithms at [math]z=1[/math] and [math]z=2[/math]

so that the logarithms no longer exist.

 

Oops, I proved it again! :note::note2:

 

You see, the properties of logarithms themselves require that

both the Beal Conjecture and Fermat's Last Theorem be true!

 

Thus, no "step by step argument" is required, because the properties of logarithms

automatically and inherently don't allow [math]x, y[/math] and [math]z[/math] all greater than 2.

 

In fact, post #21 (the Craig D Hypography version of the proof)

shows that if the problem is stated using logarithms to prevent [math]T=1[/math],

(which should be prevented since unity can not be defined in terms of itself

and can not be a non-trivial common factor) then the problem practically solves itself!

 

It is, in fact, a real "mindblower" when you realize that the solution to

the worlds most famous math problem is nothing more than

a hitherto undiscovered property of logarithms!

 

This is only one reason why I honestly believe, to the core of my being,

that my "Blazys terms" deserve further investigation.

In fact, it's a tragedy that they are so poorly understood.

 

Come on, when all is said and done, isn't it wonderfull that in the end,

it turns out that both the Beal Conjecture and Fermat's Last Theorem

can be demonstrated as being true using only "their own terms"

rather than some very, very, very distantly related concepts and constructs

involving "modular forms", "eliptical curves" and even "imaginary numbers"?

 

Shouldn't such good news be spread rather than "swept under a rug" ?

Isn't science better served when self evident truth is not deemed "silly" ?

 

Don.

[Don Blazys]

It's hard to be taken seriously

when proposing a solution to a famous problem.

"Red flags" go up immediately, people become very incredulous,

and almost uniformly adopt a "who does he think he is?" attitude.

You must then walk an impossibly fine line

so as not to appear either rude or patronizing.

Hurt feelings and bruised egos are practically unavoidable,

and thus your idea is bound to wind up in the "silly section",

regardless of how "thought provoking" it really is.

It's just the "nature of this beast", so I don't blame anyone,

exept perhaps myself, for not being a little more "tactfull".

 

Don.

[Don Blazys]

Thanks Qfwfq, for moving this thread back to where it belongs!

 

There is absolutely no logic in anger,

so please don't take to heart anything that I may have written in anger.

 

Besides, if this idea were to fail,

then I would be the first to request that it be moved back to the "silly section"!

 

For now, I'm simply overjoyed :) that this proof is again in a place

where it can be reviewed and scrutinized by top notch mathematicians,

some of whom may have the "power" or "clout" to make a difference.

 

You just never know.

 

Don.

[Don Blazys]

By now, some of you may be asking: "What is Don after? What does he really want?"

 

Well,

 

All I want is for a top notch mathematics department

at some prestigious college or university

to give me a conclusive answer, in writing,

as to the question of whether or not my proof is correct.

 

So far, all I've got are "inconclusive" letters and e-mails

with phrases such as "some support",

(which can be construed as meaning "no full support"),

and sentences such as: "We decline to publish many papers whose correctness is not in question".

(which can be construed as meaning "not necessarily your paper !".)

 

I also have other letters that say things like:

"I have not been able to find a major flaw in Don's proof",

(which may be interpreted as "a flaw may exist, but I haven't found it"),

and "He may be right",

(which of course can also mean "he may be wrong!").

 

Thus, what I really need is a letter from a top notch math department

at some prestigious college or university

that says something "unequivocal" like:

"Don's proof is simply a natural consequence of logarithms

and must therefore be correct".

 

Does anyone have any suggestions as to how this may be accomplished?

Perhaps someone here knows a mathematician in a top notch math dept.

and can invite them to Hypography for an discussion on this topic.

 

Don.