Close! Yes I have to admit…I was mistaken, I have no exception to present in order to support my argument. But, I was thinking of this*:
10[sup]2[/sup] + 11[sup]2[/sup] + 12[sup]2[/sup] = 13[sup]2[/sup] + 14[sup]2[/sup]
Someone at work pointed this out to me within the last year, and I found the relationship so intriguing, it was still kinda fresh in my mind.
*I didn’t catch the criteria: n>2
Thanks all for your replies!
I really hate it when people nitpick about pointless things, but more as a point of interest than anything else it should be criterion, as criteria is actually a plural.
It does not fit the schema x^n + y^n = z^n either.
Even without the the n>2 condition, you were on the wrong track with your example. It’s x[sup]n[/sup]+y[sup]n[/sup] <>z [sup]n[/sup], not x[sup]n[/sup]+y[sup]n[/sup]+v[sup]n[/sup] <>z [sup]n[/sup]+w[sup]n[/sup]. It helps to learn how to read math notation.
Indeed. If n = 2 solutions are as common as dirt. It is because they are so common for n = 2 that makes the fact that they are completely absent for n > 2 interesting.
Also: 3[sup]1[/sup] + 4[sup]1[/sup] = 7[sup]1[/sup] and 1[sup]2[/sup] + 1[sup]2[/sup] + 1[sup]2[/sup] + 1[sup]2[/sup] = 2[sup]2[/sup]. None of which have anything to do with the Google logo (i.e., Fermat’s last theorem).
That snarkiness having been said, if you want to understand the phenomenon you find so intriguing, it’s actually quite simple: Put another way, the relationship is that 12[sup]2[/sup] = [(12 + 1)[sup]2[/sup] - (12 - 1)[sup]2[/sup]] + [(12 + 2)[sup]2[/sup] - (12 - 2)[sup]2[/sup]]. Put even another way, the relationship is that 12 = ([(12 + 1)[sup]2[/sup] - (12 - 1)[sup]2[/sup]] + [(12 + 2)[sup]2[/sup] - (12 - 2)[sup]2[/sup]])/12.
In other words, the phenomenon is nothing more than that X = ([(X + 1)[sup]2[/sup] - (X - 1)[sup]2[/sup]] + [(X + 2)[sup]2[/sup] - (X - 2)[sup]2[/sup]])/X has a nice integer solution X = 12. But of course it does: each [(X + k)[sup]2[/sup] - (X - k)[sup]2[/sup]]/X simplifies to 4k, so the equation simplifies to X = 4 * (1 + 2).
Indeed, by the same reasoning, we have more generally that for any list of numbers, there is a unique nonzero solution to A[sup]2[/sup] + sum of (A - each element of the list)[sup]2[/sup] = sum of (A + each element of the list)[sup]2[/sup], that unique solution being 4 * the sum of the list. So, for example, with 4 * (1 + 2 + 3), we get 21[sup]2[/sup] + 22[sup]2[/sup] + 23[sup]2[/sup] + 24[sup]2[/sup] = 25[sup]2[/sup] + 26[sup]2[/sup] + 27[sup]2[/sup]; with 4 * (1 + 10 + 100 + 1000), we get 3444[sup]2[/sup] + 4344[sup]2[/sup] + 4434[sup]2[/sup] + 4443[sup]2[/sup] + 4444[sup]2[/sup] = 4445[sup]2[/sup] + 4454[sup]2[/sup] + 4544[sup]2[/sup] + 5444[sup]2[/sup]; and so on.
Well, alright, there isn’t any nonzero solution if the list sums to zero.
So what I should have said is: For any list of numbers, the solutions are precisely zero and 4 * the sum of the list. (Zero is always a solution, for any list of numbers, but not a very intriguing one…)
First off, let me make it clear that Google did have on the page “[n > 2]”. So it was correct as stated. But did you know that 3^3 + 4^3 + 5^3 = 6^3? This is interesting for a couple of reasons. One is that Hardy somewhere claims that Ramanujan’s famous statement was the 1729 is the least positive number that is the sum of two cubes in two different ways. Note the omission of positive cubes. So 91 = 3^3 + 4^3 = (-5)^3 + 6^33 is actually the smallest such positive integer. In addition, 8*91 = 728 = 6^3 + 8^3 = (-10)^3 + 12^3 = 9^3 + (-1)^3 is a sum of three cubes in three different ways.
I just have to bring this up, because it keeps bothering me each time I see the thread title:
I find the OP confusing, because it mentions a proof (“the Google math proof”). Yesterday’s Google page showed us a theorem (Fermat’s Last Theorem) but said nothing about a proof of that theorem. So I’m really confused as to what “proof” the OP is referring to, though I suspect he’s confusing “proof” with “theorem”; and that what he refers to as a “hole in the proof” is really a counterexample to the theorem.
It’s true that a counterexample, if one were to be found, would mean that the theorem is false. And that, in turn, would mean that any supposed proof of the “theorem” would have to be invalid—to have a hole in it somewhere, a flaw in the logic. But, as we’ve seen, the counterexample(s) that the OP had in mind don’t really contradict the theorem as correctly understood.
It’s also possible for a purported proof of a true theorem to have a hole in it, a gap in the logic where something doesn’t necessarily follow from something else. This would mean that the proof was invalid (unless the flaw could be fixed), but it wouldn’t tell us anything one way or another about the truth or falsity of the theorem.
We don’t have Fermat’s own “proof”—the one he claimed was too long to fit in the margin. We can speculate on what he had in mind, and it’s extremely likely that the proof he had in mind did indeed have a flaw. If he truly did have a valid proof, no one else has been able to come up with what it could have been; and anyone who could do so would become world-famous (in the mathematical world at least).
The proof we do have, from Andrew Wiles in the 1990s, is far more advanced, using mathematical techniques that Fermat would have had no way of knowing about. Wiles’s proof, as it was originally presented, did have a flaw, a hole, which would have prevented it from being considered a valid proof, but Wiles was able to fix the hole in his proof.
To expand on that, it’s even possible for a statement to be completely true, but yet for there to be no valid proof for it at all. Further, there are also statements which don’t even have a well-defined truth value, but which can equally well be true or false.
Well, where “valid proof” is taken relative to some fixed particular formal system or another… Every statement is provable in many formal systems, and disprovable in many formal systems. It’s just a matter of which particular mathematical language-game/context one is interested in.
For example, “There exists an x such that x * x = 1 + 1” cannot be proven (or disproven) from the axioms for an ordered field; this is just as archetypal an example of independence as any Goedel-type phenomena. But of course this statement has many proofs and disproofs, relative to various further systems: e.g., it can be proven given the continuity principle that every polynomial’s range is convex, and it can be disproven given the induction principle that any first-order property which holds of 0 and 1 and is closed under +, -, *, and / holds universally.
There’s no such thing as universal unprovability; just unprovability relative to particular proof systems.
If one is willing to take this tack (and I certainly am), then one ought probably not consider any statements mathematically true except relative to the contexts in which they are provable. Which would obviate the idea of true but unprovable statements.
In other words, what is the difference, for you, between alleged true but unprovable mathematical statements and alleged neither true nor false mathematical statements?
This is an interesting fact, but I think it’s unwarranted pedantry to be harsh on Hardy’s wording: one might just as well note that he didn’t explicitly forbid cubes of arbitrary real numbers, so that he was “wrong” on account of 1[sup]3[/sup] + 3[sup]3[/sup] = 2[sup]3[/sup] + 2.7144…[sup]3[/sup]. I think it’s perfectly fair to allow that “cube” can be used to mean “cube of a natural number”, instead of having to implicitly mean “cube of an integer” or some such thing…
Actually, I’m no longer sure such a weak (i.e., first-orderized) induction principle suffices for the disproof (it might, for all I know, but the precise machinery required to establish the requisite properties for the Robinson first-order definition of the integers within the field of rationals isn’t quite as clear as I initially thought). At any rate, a suitably second-order version of the same induction principle gets the job done straightforwardly.
[Not that it really matters, so far as the general idea the example was meant to illustrate goes. Indeed, not that correcting the mistake was probably burning on anyone’s mind but mine own. :)]
Quoth Indistinguishable:
Properly, one ought not to consider any mathematical statement at all, let alone considering it to have any value, absent some context for the statement (and yes, I know that at some point this must regress to “you know what I mean” when specifying the context).
Well, the standard example of a statement which can be either true or false is the continuum hypothesis, under the Zemelo-Fraenkel axioms. Your example of the existence of the square root of 2 under the field axioms is another.
As for something that’s true but unprovable, the halting problem (or any number of problems reducible to it) can work, there. If you specify a Turing machine and a program for it, then that program will either halt, or it won’t. If the statement that a particular program will run forever is false, then it’s trivial (though time-consuming) to prove it false, just by running the program until it does halt. And if the statement can be proven false, then it cannot be true. So, while a statement of the form “this program will run forever” can be true but unprovable, it cannot be of the completely indeterminate form where it being true or false is either one consistent with the axioms.
But I’m sure that you anticipated an answer like that, and have some mathematical gotcha waiting for me.
Well, what I meant was, for example, it’s perfectly cromulent to take simply the field axioms as one’s context, without specifying anything further; it’s not a context which says “Yes” or “No” to everything, but it’s still a perfectly cromulent context. And one wouldn’t consider a statement mathematically true in that context except to the extent that it might be provable in that context. The existence of a square root of 2 is not true in the context of the field axioms alone, nor is the non-existence of a square root of 2, though each of these is true relative to some other contexts. Everyone is happy to accept that that’s the way it is.
And similarly, it’s perfectly cromulent to talk about the ZF axioms, without specifying any further context than that; yet one might(/ought) not consider a statement mathematically true in the context of that particular mathematical game except to the extent that it might be provable in that game. The continuum hypothesis is not true in the context of the ZF axioms alone (nor is its negation), though it is true relative to some other contexts. Quite a few people are happy to accept that that’s the way it is.
And similarly…
My only “gotcha” is that this is the equivalent of saying “Either there is a square root of 2 or there isn’t” or “Either the continuum hypothesis is true or it’s false”. There’s no reason to suddenly switch attitudes.
I am just as willing to say there’s no truth of the matter as to whether such-and-such a program P runs forever except relative to particular proof systems which say “P runs forever” as I am for any of the other things; that is, I consider “P runs forever” a context-dependent statement as well. In the context of this system for proving that things run forever, it is mathematically true that P runs forever, in the context of that system, maybe not. That sort of thing.
There is the idea that because we have set out standard rules for what is to directly, unambiguously count as P halting, that therefore we have set out just as unambiguously the conditions for what counts as P never halting. But this is not so; the incompleteness phenomena is the very observation that we haven’t done so. We may well find that some rule-systems of interest readily tell us “P never halts” while struggling and failing to produce such a statement in other rule-systems. And that’s alright; it’s just the context-sensitivity of the statement “P never halts”.
If one wants to say “But either P really does run forever or it really doesn’t; there’s some external truth of the matter out there” in some Platonic, God’s-eye sense, one can go ahead and say it, but it doesn’t do anything to say it. It’s like saying “Either Sherlock Holmes really has blood type AB or he really doesn’t; there’s some external truth of the matter out there, even if we can’t find it in any of the stories” or “Perhaps there are secret conditions under which pawns may legally move backwards in chess, unbeknownst to the world chess community” or such things.
To be mathematically true, as far as I’m concerned, is to follow by the rules one sets out; that is, to be provable (in whatever system). One can imagine some other concept of truth if one likes, but it has no relevance to anything; an unobservable fact affects nothing. Besides, a mathematical entity is not some physical entity out in the world; it’s a description of an abstract pattern, a figure in some abstract game, and has all and only the properties we imbue it with by the rules we set out. (Useful in describing the world, yes, because if a game’s rules are simple enough, one may play it without intending to. But the usefulness of the games we study in mathematics oughtn’t lead us to suppose them to have mystical secret rules beyond the ones we defined them with.)