A question re Fermat’s Last Theorem

[SLD]

OK, so we know that there are no integers x, y, z, which satisfy the equation z³=x³+y³

But what about w, x, y, z, integers and w³=x³+y³+z³?

Are these still considered Diophantine equations?

 

[Bomb#20]

OK, so we know that there are no integers x, y, z, which satisfy the equation z³=x³+y³

But what about w, x, y, z, integers and w³=x³+y³+z³?

12³=10³+9³+(-1)³

Are these still considered Diophantine equations?

Yup.

 

[SLD]

OK, so we know that there are no integers x, y, z, which satisfy the equation z³=x³+y³

But what about w, x, y, z, integers and w³=x³+y³+z³?

12³=10³+9³+(-1)³

Are these still considered Diophantine equations?

Yup.

OK so I wonder if a general rule follows from FLT that there are no integer solutions to such equations for n+1, n being the number of variables in the equation.

i.e.

zⁿ=SUM{x_iⁿ} from i = 1 to n-1

Then there are no integer solutions.

If there is no proof of this theorem, I’d like it to be called SLD’s last theorem.

 

[Swammerdami]

The one I like, because its elegant (3,4,5;6) structure mimics the elegant Pythagorean (3,4;5), is
3^3 + 4^3 + 5^3 = 6^3

12³=10³+9³+(-1)³


Yup.

OK so I wonder if a general rule follows from FLT that there are no integer solutions to such equations for n+1, n being the number of variables in the equation.

i.e.

zⁿ=SUM{x_iⁿ} from i = 1 to n-1

Then there are no integer solutions.

If there is no proof of this theorem, I’d like it to be called SLD’s last theorem.

Here's a counterexample to your conjecture:
95800^4+217519^4+414560^4=422481^4
Anyway, I think you'll find you were beaten to this conjecture ... by an obscure mathematician named Leonhard Euler!

 

[SLD]

The one I like, because its elegant (3,4,5;6) structure mimics the elegant Pythagorean (3,4;5), is
3^3 + 4^3 + 5^3 = 6^3

12³=10³+9³+(-1)³


Yup.

OK so I wonder if a general rule follows from FLT that there are no integer solutions to such equations for n+1, n being the number of variables in the equation.

i.e.

zⁿ=SUM{x_iⁿ} from i = 1 to n-1

Then there are no integer solutions.

If there is no proof of this theorem, I’d like it to be called SLD’s last theorem.

Here's a counterexample to your conjecture:
95800^4+217519^4+414560^4=422481^4
Anyway, I think you'll find you were beaten to this conjecture ... by an obscure mathematician named Leonhard Euler!

Dang. So much for my immortality as a mathematician.

BTW, how the fuck did you find the numbers to the fourth power above?

 

[Swammerdami]

The one I like, because its elegant (3,4,5;6) structure mimics the elegant Pythagorean (3,4;5), is
3^3 + 4^3 + 5^3 = 6^3

Here's a counterexample to your conjecture:
95800^4+217519^4+414560^4=422481^4
Anyway, I think you'll find you were beaten to this conjecture ... by an obscure mathematician named Leonhard Euler!

BTW, how the fuck did you find the numbers to the fourth power above?

The abstract is available on-line. The full paper (from 1988) is yours for just $33, or $14.95 if you are an IEEE member.

Abstract:
The smallest counterexample to Euler's generalization of Fermat's Last theorem is 95800/sup 4/+217519/sup 4/+414560/sup 4/=422481/sup 4/. The author explains how this solution was found by an exhaustive data-parallel search on several Connection Machine systems. An outline of the history of the problem and the architecture of the data-parallel Connection Machine system are presented along with the parallel algorithm.

I Googled to look for a free copy of the paper. No luck, but saw an earlier-discovered simpler-number counterexample, though for N=5 instead of N=4:

27^5 + 84^5 + 110^5 + 133^5 = 144^5

 

[Swammerdami]

OK, so we know that there are no integers x, y, z, which satisfy the equation z³=x³+y³

But what about w, x, y, z, integers and w³=x³+y³+z³?

12³=10³+9³+(-1)³

Shame on you! You know Fermat's Last Theorem specifies positive integers, whether SLD said so explicitly or not. If you're allowing negatives, why not allow zero as well, producing lots of exceptions, e.g.
69^5 = 69^5 + 0^5 + 0^5

Your equation can be re-arranged to show the famous Ramanujan  Taxicab number
1729 = 10^3 + 9^3 = 12^3 + 1^3

If we allow this formula (sum of two powers equals sum of two powers) instead of the usual (power equals sum of three powers), there are much simpler N=4 exceptions than the one I showed earlier:
134^4 + 133^4 = 158^4 + 59^4
227^4 + 157^4 = 239^4 + 7^4
257^4 + 256^4 = 292^4 + 193^4

 

[Bomb#20]

OK, so we know that there are no integers x, y, z, which satisfy the equation z³=x³+y³

But what about w, x, y, z, integers and w³=x³+y³+z³?

12³=10³+9³+(-1)³

Shame on you! You know Fermat's Last Theorem specifies positive integers, whether SLD said so explicitly or not.

What the heck difference does it make whether Fermat's Last Theorem specifies positive integers? With or without the restriction, it's the same problem! x² = (-x)²

If you're allowing negatives, why not allow zero as well, producing lots of exceptions, e.g.
69^5 = 69^5 + 0^5 + 0^5

You appear to have answered your own question. We don't allow zero because allowing zero stops it from being an interesting problem.

Your equation can be re-arranged to show the famous Ramanujan  Taxicab number
1729 = 10^3 + 9^3 = 12^3 + 1^3

Which of course is where I got that solution from.

If we allow this formula (sum of two powers equals sum of two powers) instead of the usual (power equals sum of three powers), there are much simpler N=4 exceptions than the one I showed earlier:
134^4 + 133^4 = 158^4 + 59^4
227^4 + 157^4 = 239^4 + 7^4
257^4 + 256^4 = 292^4 + 193^4

Sounds good to me. Welcome to the dark side.

 

[Tigers!]

If there is no proof of this theorem, I’d like it to be called SLD’s last theorem.

Wanting a last theorem implies that you have at least one other theorem. What is it?

 

[Keith&Co.]

If there is no proof of this theorem, I’d like it to be called SLD’s last theorem.

Wanting a last theorem implies that you have at least one other theorem. What is it?

Orrrrrrrrrrr, if you die in the middle of resolving it, it's DEFINITELY your Last.

 

[Swammerdami]

The great mathematician Godfrey Hardy had a "last theorem" lined up, at least according to a story he liked to tell.

About to cross the North Sea by boat and knowing there was some risk of foundering, he sent a letter home to England: "I have proved the Riemann Hypothesis!" He had no proof but decided that if he were to die, he should die with great fame!