Improving On Euler's Conjecture

[David Sklansky]

He was wrong when he said that three fourth powers could never add up to a fourth power.

But what about three sixth powers adding up to a sixth power?

This post is related to an idea that has been kicking around in my head for forty years. I made a misguided effort to introduce this idea with a pevious post about Fermat's Theorem. I'll try again using the above two conjectures. Without going into too much detail I will say that my idea separates number theory questions into two categories. The above conjectures are in different categories. As a hint to what I am driving at, if I conjectured that EIGHT sixth powers could never add up to a sixth power it would be a question in the same category of Euler's original and wrong conjecture. (My guess is that this third conjecture is wrong also. But not the second one.)

[wheeler]

Your "idea" is sufficiently vague that it's almost certainly correct. Or incorrect. How can we tell? You have to be more precise. Mathematicians don't usually explain ideas by not going into "too much detail" and only giving "hints" to what they're trying to say.

For example, from what you've written above I might conclude that you want to just divide conjectures into two categories: those that are right and those that are wrong! But that's certainly not what mean.

Couldn't you try to explain what your two categories of conjecture are, so that we can help you massage your meta-conjecture into a form that has an answer?

- wheeler

[David Sklansky]

For now I'd like to not muddy the waters and merely see what mathmeticians have to say about the second conjecture.

[mslif]

I am trying to follow what you are saying (hopefully I am not completely off, if so please be kind):

I do not think that it will be in the same cathegory because your are talking about:
a^6+b^6+c^6+d^6+e^6+f^6+h^6+i^6=j^6
Euler's Conjecture is:
a^4+b^4+c^4=e^4

Euler proposed that for every integer greater than 2, the sum of n – 1 nth powers of positive integers cannot itself be an nth power.

The fact that you are conjuncturing EIGHT sixth powers does not follow Euler's theory.

[wheeler]

[ QUOTE ]
For now I'd like to not muddy the waters and merely see what mathmeticians have to say about the second conjecture.

[/ QUOTE ]

The second conjecture is open.

The third conjecture is false. See
http://mathworld.wolfram.com/Diophan...6thPowers.html
for counterexamples.

[PairTheBoard]

[ QUOTE ]
Your "idea" is sufficiently vague that it's almost certainly correct. Or incorrect. How can we tell? You have to be more precise. Mathematicians don't usually explain ideas by not going into "too much detail" and only giving "hints" to what they're trying to say.

For example, from what you've written above I might conclude that you want to just divide conjectures into two categories: those that are right and those that are wrong! But that's certainly not what mean.

Couldn't you try to explain what your two categories of conjecture are, so that we can help you massage your meta-conjecture into a form that has an answer?

- wheeler

[/ QUOTE ]

From David's Fermat Thread, I believe his metaconjecture goes something like this:

"There exist True Number Theory Statements that are Impossible to prove."

I think this goes beyond Godel who showed that within any logical system rich enough to include the integers there will exist statements that will be impossible to prove or disprove from within that logical system. David asserts there will be such statements that are in fact true, yet impossible to prove.

I'm really not sure what to make of his conjecture myself. I'm a bit bothered by what his use of the term "true" means.

PairTheBoard

[PairTheBoard]

The Sklansky Metaconjecture:
"There exist True Number Theory Statements that are Impossible to prove."


One thing that seems clear is that the Sklansky Metaconjecture is impossible to prove by exhibiting an actual example. But since it's an "existence" type statement its proof really doesn't require an actual example, only a proof that one exists.

PairTheBoard

[PairTheBoard]

[ QUOTE ]
The Sklansky Metaconjecture:
"There exist True Number Theory Statements that are Impossible to prove."


One thing that seems clear is that the Sklansky Metaconjecture is impossible to prove by exhibiting an actual example. But since it's an "existence" type statement its proof really doesn't require an actual example, only a proof that one exists.

PairTheBoard

[/ QUOTE ]

Also, even if the Sklansky Metaconjecture could be proven and thus the Skansky Class of number theory conjectures shown to be non empty, it would be impossible to ever show that a particular number theory conjecture actually fell in the Sklansky Class.

PairTheBoard

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
The Sklansky Metaconjecture:
"There exist True Number Theory Statements that are Impossible to prove."


One thing that seems clear is that the Sklansky Metaconjecture is impossible to prove by exhibiting an actual example. But since it's an "existence" type statement its proof really doesn't require an actual example, only a proof that one exists.

PairTheBoard

[/ QUOTE ]

Also, even if the Sklansky Metaconjecture could be proven and thus the Skansky Class of number theory conjectures shown to be non empty, it would be impossible to ever show that a particular number theory conjecture actually fell in the Sklansky Class.

PairTheBoard

[/ QUOTE ]


I suspect where David would want to go from that point is to develop some kind of theory that would produce "probabilities" of Number Theory Conjectures falling in the Sklansky Class.


He might then open a world wide online Casino to take bets on the action. Unfortunately the only bets that could ever be won would be those against a conjecture belonging to the Sklansky Class.

PairTheBoard

[jason1990]

[ QUOTE ]
From David's Fermat Thread, I believe his metaconjecture goes something like this:

"There exist True Number Theory Statements that are Impossible to prove."

I think this goes beyond Godel who showed that within any logical system rich enough to include the integers there will exist statements that will be impossible to prove or disprove from within that logical system. David asserts there will be such statements that are in fact true, yet impossible to prove.

I'm really not sure what to make of his conjecture myself. I'm a bit bothered by what his use of the term "true" means.

PairTheBoard

[/ QUOTE ]
I don't think it goes beyond Godel. Suppose the conjecture

"There do not exist three sixth powers adding up to a sixth power."

is impossible to prove or disprove. Then it is true. Why? Well, suppose it is false. Then there exists three sixth powers adding up to a sixth power. But if they exist, then there exists a finite proof that this conjecture is false. But that cannot be, because we assumed the conjecture was impossible to disprove. Therefore, it is impossible to prove that this conjecture is impossible to prove or disprove, because that would be a proof of the conjecture. But there may be statements of this form which are impossible to prove or disprove, and whatever statements those are, they are true.

From "Goedel's Proof" by Ernest Nagel and James R. Newman: Goedel showed "that there is an endless number of true arithmetical statements which cannot be formally deduced from any given set of axioms by a closed set of rules of inference. It follows that an axiomatic approach to number theory, for example, cannot exhaust the domain of arithmetical truth."