On the road to prove Fermat's Impossiblity Equation !

[Commander]

Any Positive Natural Number N can be written as a positive sum of the Powers of 2 !

 

N = 2^a + 2^b ..... with as many terms as require where a , b , c etc are all Natural positive Numbers from [ 0,1,2,3 .....]

 

These powers a, b, etc in the sequence will be PRESENT or ABSENT in the Equation only once.

 

No repetition is required.

 

This is nothing but a Binary Number representing the powers of 2 which add up to be equal to N.

 

Similarly any N can be written as N = an additions or subtractions of Powers of 3 OCCURRING only once.

 

N = 3^a ± 3^b ± 3^c etc again a,b,c etc needing to appear only once !

 

Any number higher than 3 will not be able to produce such a Sum with their Powers Occurring only once !

 

Not possible to cover all N by 4^a ± 4^b ± 4^c ....... or similarly by the powers of 5,6,7 ... or such higher numbers.

[fiveworlds]

Any Positive Natural Number N can be written as a positive sum of the Powers of 2 !

 

Uh huh. 1 boss 2^1*5=10, 10/10=1

 

Fermat is right you know. no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two other than 1

Edited December 22, 2014 by fiveworlds

[Commander]

Uh huh. 1 boss 2^1*5=10, 10/10=1

 

Fermat is right you know. no three positive integers a, b, and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two other than 1

 

I think you misunderstood what I said.

 

I know what Fermat's stated what you are quoting.

 

That was his statement which needs to be proved or disproved.

[imatfaal]

Thomas: it has already been proved - although Andrew Wiles' proof is a real monster and cannot be an analogue to that which Fermat claimed.

[Acme]

Thomas: it has already been proved - although Andrew Wiles' proof is a real monster and cannot be an analogue to that which Fermat claimed.

Correct. We can't know if Fermat was lying or not, and as you say, if he wasn't lying then his proof was not Wiles' proof.

 

So Commander, do you think you have a proof, or are you looking to tease one out of a group effort here?

[Commander]

Thomas: it has already been proved - although Andrew Wiles' proof is a real monster and cannot be an analogue to that which Fermat claimed.

 

Correct. We can't know if Fermat was lying or not, and as you say, if he wasn't lying then his proof was not Wiles' proof.

 

So Commander, do you think you have a proof, or are you looking to tease one out of a group effort here?

 

Well , at this stage I am working on my own in both Mathematical and Cosmology Fields to give wings to my own thoughts and not get restricted by existing Theories.

 

Or else it will suppress my own line of thinking and intuitiveness.

 

Sometimes I try and bring in my thinking in solving some Puzzles [where invariably heavy thinking is needed] into possible Generalization or Theories.

 

For example We used a TRINARY SEARCH Algorithm in solving 39 Balls Problem better than a Binary Search Method.

 

Similarly, here again the Property exhibited by Number 3 I noted from the Solution of another interesting Puzzle.

 

We all know that 0 , 1 , 2 and 3 are the Bricks of Number Theories and 3 is on the Wall Dividing 0,1,2 and the rest 4.5.6 etc..

 

Like Fermat's theorem defines what is Possible with the Power 2 but not Possible for Powers 3 or more !

 

At this point I am looking at a possible link arising.

 

Also if I add Walker's Equation too I can state :

 

Every Possible Rational number can be expanded into a Sum of Powers of 2 Occurring Only Once and some Elements of the Matrix of Walker's Equation - perhaps occurring only once !

[ajb]

Or else it will suppress my own line of thinking and intuitiveness.

The problem is that you should not forget everything that we have learned and discovered before tying to contribute; you need to have the tools at hand. It can also mean that you are working on things either disconnected from the community or things that are considered uninteresting, maybe even solved already.

Or else it will suppress my own line of thinking and intuitiveness.

The problem is that you should not forget everything that we have learned and discovered before tying to contribute; you need to have the tools at hand. It can also mean that you are working on things either disconnected from the community or things that are considered uninteresting, maybe even solved already.

[Acme]

The problem is that you should not forget everything that we have learned and discovered before tying to contribute; you need to have the tools at hand. It can also mean that you are working on things either disconnected from the community or things that are considered uninteresting, maybe even solved already.

 

...

Just because something is solved/proved does not make a new or different solution/proof uninteresting. There must be dozens of different proofs of the Pythagorean theorem and new ones are still of interest. You commented in one of Commander's other threads:

That is cool.

It is a nice result and one that is not hard to prove. However, simply being correct is not enough for it to be interesting. Good luck with finding applications.

A math result that has no [known] applications is not of necessity uninteresting. Witness the popularity of books on 'recreational' mathematics. Pure mathematics is no less useful to the human intellect than applied mathematics is to those endeavors to which it is applied. Thinking is to the mind as exercise is to the body.

[Commander]

Just because something is solved/proved does not make a new or different solution/proof uninteresting. There must be dozens of different proofs of the Pythagorean theorem and new ones are still of interest. You commented in one of Commander's other threads:

A math result that has no [known] applications is not of necessity uninteresting. Witness the popularity of books on 'recreational' mathematics. Pure mathematics is no less useful to the human intellect than applied mathematics is to those endeavors to which it is applied. Thinking is to the mind as exercise is to the body.

 

Good Points ! +1

[ajb]

We run the risk of being off topic here...

 

Just because something is solved/proved does not make a new or different solution/proof uninteresting. There must be dozens of different proofs of the Pythagorean theorem and new ones are still of interest.

New proofs can be interesting, especially shorter proof or those that make unexpected links between different branches of mathematics.

 

But I have not yet read a paper on a new proof of the Pythagorean theorem.

 

The point is you don't really want to tread over old ground all the time.

 

A math result that has no [known] applications is not of necessity uninteresting.

I agree, and never wished to give the impression I thought that. However, the specific result in the other thread is an infinite sum that can be shown to converge to 1 using two well known sums and a little algebra.

 

Also, by applications I also mean applications in pure mathematics. Results are always more interesting of there is some motivation for them, and of course this motivation may be very theoretical or it could be more practical. Either way doing something because you can is not always enough.

 

Witness the popularity of books on 'recreational' mathematics.

Okay, and maybe this kind of mathematics should be in that sector.

 

Pure mathematics is no less useful to the human intellect than applied mathematics is to those endeavors to which it is applied.

Agreed, again I am sorry if you ever got the impression that I though otherwise.

 

So back on topic... as already stated Fermat's Last Theorem is now proved in several stages with Wiles putting it all together. A shorter direct proof would be great, but I think it would be too much to ask for. The statement is simple but it requires a lot of heavy duty mathematics to solve.

Edited December 24, 2014 by ajb

[Dekan]

Thomas: it has already been proved - although Andrew Wiles' proof is a real monster and cannot be an analogue to that which Fermat claimed.

Andrew Wiles's proof, does seem too monstrously long to be satisfactory. Fermat claimed to have a simpler proof. Even if it couldn't be written in the margin. Has any one here, even the best mathematicians on the board, fully read through all the hundreds of pages of the Wiles proof

 

Probably not. The proof has been accepted, as an act of faith.

[imatfaal]

Andrew Wiles's proof, does seem too monstrously long to be satisfactory. Fermat claimed to have a simpler proof. Even if it couldn't be written in the margin. Has any one here, even the best mathematicians on the board, fully read through all the hundreds of pages of the Wiles proof

 

Probably not. The proof has been accepted, as an act of faith.

 

No - but I know (from another forum) one of the men who checked it. Whilst the paper was undergoing rigorous peer-review (note not faith-based) he was rumoured to have found a counter-example (this was quite possible as he was a numerical computation specialist) and a panickly friend of Wiles phoned up hoping to scotch the rumour - luckily it was just a sillly internet meme.

 

Science is a massively interconnect web of relationships - we know who we trust to read a paper as complicated as Wiles. I am lawyer - not a mathematician but I know that I could post a question here and ask people like AJB who would understand so so much more than me about Wiles, AJB could ask colleagues who were completely cognizant of the whole area, and AJB or these colleagues I mentioned could get on the blower to Wiles. It is not a matter of faith - it is a matter of science.

 

And if you think trust is the same as faith - then Phi will set you right as he has posted on this at length and it is worth searching for a reading; and if that doesn't convince you think of basic human nature - the mathematician that showed a flaw in Wiles post would be headline news on real news programmes and be on a gravy train for life

[Dekan]

 

No - but I know (from another forum) one of the men who checked it. Whilst the paper was undergoing rigorous peer-review (note not faith-based) he was rumoured to have found a counter-example (this was quite possible as he was a numerical computation specialist) and a panickly friend of Wiles phoned up hoping to scotch the rumour - luckily it was just a sillly internet meme.

 

Science is a massively interconnect web of relationships - we know who we trust to read a paper as complicated as Wiles. I am lawyer - not a mathematician but I know that I could post a question here and ask people like AJB who would understand so so much more than me about Wiles, AJB could ask colleagues who were completely cognizant of the whole area, and AJB or these colleagues I mentioned could get on the blower to Wiles. It is not a matter of faith - it is a matter of science.

 

And if you think trust is the same as faith - then Phi will set you right as he has posted on this at length and it is worth searching for a reading; and if that doesn't convince you think of basic human nature - the mathematician that showed a flaw in Wiles post would be headline news on real news programmes and be on a gravy train for life

Thanks imatfaal.

I just feel an instinctive doubt about any "proof" of a simple proposition - if the proof takes several hundred pages of esoteric maths. The maths might contain an error somewhere. Probably at the beginning, in accordance with Sod's Law!

[imatfaal]

Thanks imatfaal.

I just feel an instinctive doubt about any "proof" of a simple proposition - if the proof takes several hundred pages of esoteric maths. The maths might contain an error somewhere. Probably at the beginning, in accordance with Sod's Law!

 

I quite agree with the first point - I like even less a proof which, to an extent works through the principle of exhaustion especially if a computer is the only thing that could realistically check; the maths might well contain an error in the first, second, third draft etc... But in a publication copy of a proof of the most famous theorem in mathematics almost no chance of a mistake; it is not zero - that would be hubris, but it is as close to zero as humans can reach.

 

Read Simon Singh excellent book on this topic

[ajb]

The proof has been accepted, as an act of faith.

His two papers went through peer review and on such a big claim I am sure they were checked very carefully at that stage. Also since then many people would have read them even if just pieces. Any big mistakes would have been noticed by now.

[fiveworlds]

That was his statement which needs to be proved or disproved.

 

No I ran Math.Pow he is correct

<script>
var exponent=3;

var number=1;
while(number<100)
{
var b=Math.pow(number, exponent);
document.write(b);
document.write("<div></div>");
number=number+1;
}
</script>

That will give you the first 100 numbers you can use no two of these add to another. Not only that but if you increase the exponent then you can increase the number of whole numbers you can't add a+b+c..n=z where the number of letters is one less than the exponent. So where the exponent is 9 a^n+b^n+c^n+d^n+e^n+f^n+g^n+h^n=k^n isn't possible.

Edited December 26, 2014 by fiveworlds

[imatfaal]

No I ran Math.Pow he is correct

<script>
var exponent=3;

var number=1;
while(number<100)
{
var b=Math.pow(number, exponent);
document.write(b);
document.write("<div></div>");
number=number+1;
}
</script>

That will give you the first 100 numbers you can use no two of these add to another. Not only that but if you increase the exponent then you can increase the number of whole numbers you can't add a+b+c..n=z where the number of letters is one less than the exponent. So where the exponent is 9 a^n+b^n+c^n+d^n+e^n+f^n+g^n+h^n=k^n isn't possible.

 

Fiveworld - Fermat's last theorem cannot be proved or disproved by running a simple search because the range of inputs is unbounded. we must ask - is there no answer to the equation from the whole of the integers greater than two - you would need to test them ALL; and that is not possible. This would also apply to Thomas' other postulates above as well - one counter-example blast the theorem to pieces but no number of confirmations of the theorem are enough.

 

Wiles was able to prove that the entire set of natural numbers in this situation could be seen through an analogue to a limited of geometric states, he was then able to show what the theorem would imply to this group of states, and then show that this does not apply at any point. For your guidance - before Wiles' proof it had been checked numerically with exponents up to several million.

[Unity+]

 

Fiveworld - Fermat's last theorem cannot be proved or disproved by running a simple search because the range of inputs is unbounded. we must ask - is there no answer to the equation from the whole of the integers greater than two - you would need to test them ALL; and that is not possible. This would also apply to Thomas' other postulates above as well - one counter-example blast the theorem to pieces but no number of confirmations of the theorem are enough.

 

Wiles was able to prove that the entire set of natural numbers in this situation could be seen through an analogue to a limited of geometric states, he was then able to show what the theorem would imply to this group of states, and then show that this does not apply at any point. For your guidance - before Wiles' proof it had been checked numerically with exponents up to several million.

Though such algorithms can lead to the actual proof.

[Commander]

Still trying hard to find a possible solution to a³ + b³ = c³

 

 

 

If we have any Integer Solution to x, y, z in the above equation then

 

we have a Solution to a³ + b³ = c³ with a,b,c as integers.

Edited January 2, 2015 by Commander

[michel123456]

square root of 3 is irrational.

[imatfaal]

square root of 3 is irrational.

 

Good Spot - but it does not spell the end. If (4xy^3-x^4) is a multiple of three then you will get sqrt(3)*sqrt(3)*sqrt(something else) - and that could be integer

[Commander]

square root of 3 is irrational.

Hi,

 

Thanks.

 

It is the Cubic Root of the segment under the bar

[michel123456]

Hi,

 

Thanks.

 

It is the Cubic Root of the segment under the bar

??

I don't follow.

 

Good Spot - but it does not spell the end. If (4xy^3-x^4) is a multiple of three then you will get sqrt(3)*sqrt(3)*sqrt(something else) - and that could be integer

So one can assume 4xy^3 -x^4 is indeed a multiple of three and that reduces to prove that sqrt(something else) can be an integer*. Or the other way round.

Provided the equation is correct (I didn't check)

 

*for all multiples of 2 it gives an integer if I am not totally wrong.

Edited January 3, 2015 by michel123456

[imatfaal]

??

I don't follow.

 

Me neither.

 

So one can assume 4xy^3 -x^4 is indeed a multiple of three and that reduces to prove that sqrt(something else) can be an integer*. Or the other way round.

Provided the equation is correct (I didn't check)

 

Sorry Michel I was missing the wood for the trees. The equation cannot be correct - according to the Commander (I haven't checked) it would be a counter-example to Fermat; and Fermat has been proven. I thought you were showing why it could not be solved with integer sums - which if that sqrt hadn't been multiplied by a second sqrt you would have done

Still trying hard to find a possible solution to a³ + b³ = c³

 

Equation for Fermat.jpg

 

If we have any Integer Solution to x, y, z in the above equation then

 

we have a Solution to a³ + b³ = c³ with a,b,c as integers.

 

[latex]z=\frac{\sqrt{3} \cdot \sqrt{4xy^3-x^4}-3x^2+6x}{6x}[/latex]

 

You get real solutions when [latex]y=+/- \frac{x}{2^{2/3}}[/latex] and [latex]z=\frac{2-x}{2}[/latex]

 

And it follows - using same logic as Michel above - if y is an integer then x cannot be (as it is divided by an irrational number) or vice versa if x is an integer then y cannot be

[Commander]

 

Me neither.

 

 

Sorry Michel I was missing the wood for the trees. The equation cannot be correct - according to the Commander (I haven't checked) it would be a counter-example to Fermat; and Fermat has been proven. I thought you were showing why it could not be solved with integer sums - which if that sqrt hadn't been multiplied by a second sqrt you would have done

 

[latex]z=\frac{\sqrt{3} \cdot \sqrt{4xy^3-x^4}-3x^2+6x}{6x}[/latex]

 

You get real solutions when [latex]y=+/- \frac{x}{2^{2/3}}[/latex] and [latex]z=\frac{2-x}{2}[/latex]

 

And it follows - using same logic as Michel above - if y is an integer then x cannot be (as it is divided by an irrational number) or vice versa if x is an integer then y cannot be

 

I think that the equation indicates (4xy³ - x⁴)^1/3 rather than 3^1/2 multiplied by(4xy³ - x⁴)^1/2