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Banker Offers $1M To Solve Beal Conjecture
oxide7 writes "A Texas banker with a knack for numbers has offered $1 million for anyone who can solve a complex math equation that has stumped mathematicians since the 1980s. The Beal Conjecture states that the only solutions to the equation A^x + B^y = C^z, when A, B and C are positive integers, and x, y and z are positive integers greater than two, are those in which A, B and C have a common factor. Like most number theories, it's "easy to say but extremely difficult to prove.""
Sorry, but I promise you that the solution was very elegant.
Bert
Fermat's last theorem requires x = y = z, and argues that there is no answer. This is a generalization.
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No, Fermat's last theorem says a^n + b^n = c^n for n > 2 has no solutions. Here's it says a^x + b^y = c^z for x,y,z > 2 only have solutions when they have a common factor. Example: 3^3 + 6^3 = 3^5.
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That's essentially what Carl Friedrich Gauss said when he was challenged to prove Fermat's Last Theorem. Something on the lines of: "I have no real interest in such endeavors since I could easily put forward a multitude of propositions which one could neither prove nor disprove."
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Give a man a gun, he can rob a bank
Give a banker an algorithm, he can rob the world.
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Whats so special about this one - does it have some mathematical relevance?
Yes, it's relevance is that mathematicians don't like empirical evidence that a statement is only 99.9999% accurate; They demand 100%. And in mathematics, you can get 100%.
And just like prime numbers, fermat's last theorem, etc., an enhanced understanding of the relationships laid out by certain formulas can, and often does, lead to an enhanced understanding of the universe -- which for some strange reason, seems to have the quality of being well-described, if not completely described, by the body of knowledge known as mathematics. And by understanding the universe better, we understand ourselves, and can make our lives easier. Creating most of our modern technology requires an understanding of mathematics -- so better math means better technology.
Relevant enough for you, or do I need to resort to a beer analogy? :)
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Worse than 'a generalization': if this conjecture is true, FLT is a trivial consequence. That's a clue that Beal's conjecture is likely significantly harder than Fermat's.
Oh great, yet another bank that wants a bealout.
Table-ized A.I.
Huh. For a second there my brain thought: "FTL? Faster Than Light (travel)? WTF!"
So, being quite cynical about such things, in what way would a proof of this conjecture allow him to make more money?
Philanthropy and advancing science are good, but my first thoughts is that if someone can prove this he stands to make massive amounts of money.
You know the old jokes about rich people paying bums on the street to fight for their own amusement? Well, extend that to mathematicians.
Demanding constant attention will only lead to attention.
Only if x=y=z. For instance, somebody above suggested 3^3 + 6^3 = 3^5 (27+216=243). If we factor out the common 3, we get 3^2 + 2*(6^2) = 3^4 (9+72=81), which no longer has the right form because 72 is not a power of any number.
If x=y=z, and if A^x+B^x=C^x where A,B,C had the same greatest common factor n, then you could divide all three numbers by n^x and get a new formula (A/n)^x+(B/n)^y=(C/n)^z where A/n, B/n, and C/n had no common factor, and if Beal's conjecture is true then these numbers cannot exist if x>2. Therefore A^x+B^x=C^x has no nontrivial solution for x>2, which is Fermat's last theorem.
Assume Beal's conjecture and you have a minimal counterexample to FLT where A^x + B^x = C^x. Then A, B, C have a common prime factor p, so (A/p)^x + (B/p)^y = (C/p)^z is a smaller counterexample to FLT, which contradicts our minimality assumption.
Beer analogy, as requested: suppose you want to be 100% certain that no one is pissing a little in your beer before you drink it. You can be 99.999% sure by ordering high-quality beer in an upscale establishment and watching the bartender fill your glass --- but you've still got that nagging fear that someone in the back room, or even the brewery, may have whizzed in the keg when no one was looking. So, instead of relying on empirical likelihoods, you go and brew your own beer, from start to finish under your watchful eye, to get that 100%-guaranteed-piss-free pint. And, ultimately, humankind's fundamental knowledge and craft of beer brewing is advanced through the initial efforts of home-brew enthusiasts.
This is the same Beal who founded Beal Aerospace. Also the same Beal who challenged the world's best professional poker players to the highest one-on-one Texas Hold'Em games ever played ($100,000/$200,000 IIRC). Also, this isn't an equation to be 'solved'. It's a conjecture to be proved (or disproved).
Huh. For a second there my brain thought: "FTL? Faster Than Light (travel)? WTF!"
WTF? World Taekwondo Federation?
Every analogy breaks down somewhere (or else it wouldn't be an analogy, but the thing analogized itself). If one wishes absolute mathematical certainty for their piss-free pint, then one must be satisfied with a mathematical pint. As thus: consider a Platonic ideal beer, symbolically represented by the word "beer." Now, imagine quaffing the beer. Theoretically, this should be satisfying and delicious. If you don't consider this exercise superior to drinking an actual beer, you may just not be cut out for pure mathematics --- consider becoming a physicist instead.
I dunno. Do you have access to infinite compute cycles?
Obviously nobody has found an exception to disprove it yet. The dude wouldn't be offering a pile of money if he were just looking to disprove it...he would just funnel the money into some supercomputer time to step through an absurd amount of integers until he comes up with an exception.
The set of integers to test is big. Really big. You just won't believe how vastly, hugely, mindbogglingly big it is. I mean, you may think that Graham's number is a lot, but that's just peanuts compared to the integers.
If a conjecture could be disproven by simply throwing computational resources at the problem, chances are that it's not particularly interesting. Many open problems in number theory have known lower bounds well above anything that could possibly be tested by a computer. For example, there is no odd perfect number less than 10**1500.
We have a fairly good hunch what Fermat's actual "proof in the margin" was. I can't remember how it goes, but it falls apart because rings Z^n with n>13 are no longer Unique Factorization Domains (UFD: a ring where all numbers have a single unique prime factorization) (or something like that). The concept of something not being a UFD was unheard of at the time of Fermat. Disclaimer: it's a few years since I did Algebra, so there may be errors in this post.
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Maybe if I present it in the form of a cryptography scheme for terrorist communications...
"When information is power, privacy is freedom" - Jah-Wren Ryel
Ah, the integers are nothing. You should have seen the real interval I had on the hook last week. I tried bring it onboard using a sieve I borrowed from Eratosthenes, but the sieve was not nearly large enough. I'm telling you, the boat nearly keeled over!