Sklansky -Fermat Conjectures

[AlphaWice]

[ QUOTE ]
1)

Proof:
Choose C s.t. C^n < A^n + B^n
Put q = A^n + B^n - C^n

Then A^n + B^n = C^n + q

So there exists A, B, C, q for which the assertion is true.

Conversely, fix any C^n. Pick q = C*C^n. Then for any A^n, B^n we have A^n + B^n = C^n + q i.e. A^n + B^n = C^(n+1) i.e. A^n + B^n = (C²)^n which cannot have a solution due to Fermat's Theorem.

So for each C, there exists a q for which the assertion is false.

This proves Conjecture One, i.e. that for some but not for all q, the equation A^n + B^n = C^n + q does not have a solution.

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(c^2)^n does not equal c^(n+1).

[Abednego]

You're just trying to find out if I am really Andrew Wiles aren't you?

[gaming_mouse]

[ QUOTE ]
In other words it might be true that (A to the n) + (B to the n) can never equal (C to the n) plus (lets just say) the number 846879032 (n greater than four), yet no proof of this fact is even theoretically findable.


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This idea of truth without proof has always seemed like a contradiction to me.

I thought the whole point of axiomatic mathematics was that you replace the notion of truth with the notion of provability. That is, by definition, what is true is what can be proved.

[SheetWise]

Alice laughed. "There's no use in trying," she said: "one can't believe impossible things."

"I daresay you haven't had much practice," said the Queen. "When I was your age, I always did it for half-an-hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."

SCOTUS

[gaming_mouse]

[ QUOTE ]

This idea of truth without proof has always seemed like a contradiction to me.

I thought the whole point of axiomatic mathematics was that you replace the notion of truth with the notion of provability. That is, by definition, what is true is what can be proved.

[/ QUOTE ]

Btw, to clarify this, I don't mean that this is my own personal notion of truth. It is just my understanding of the term "truth" as it used in mathemetical contexts.

[jason1990]

I'm no logician, but my understanding is that the notion of "truth" transcends provability. In my possibly misguided perception of things, I imagine it having something to do with the fact that proofs must end. So, for instance, it may be true that statement P(n) holds for all n, but the only "proof" would be to check P(1),P(2), and so on. So although you could never prove it (in finite time), it is still true in the sense that you can never find a counterexample.