Sorry about the vagueness of the title - I didn’t convey the essence of my question very well.
In any case, reading this article about the recent discovery that infinitely many prime numbers come in ‘pairs’, reminded me of something I read many years ago, before ‘Fermat’s Last Theorem’ had been proved by Andrew Wiles.
Specifically, I remember reading an article which noted that the Theorem had been proved for values of the exponent of less than ten million (or some other large integer; the actual number is irrelevant to my question). The author went on to state, though, words to the effect that one shouldn’t become confident in its general validity just because it held for the first ten million cases. Specifically, he/she then referred to another theorem (I think also from number theory) which which failed only when n > 10[sup]10[sup]14[/sup][/sup] (or some other fantastically large number), with ‘n’ being the theorem’s key variable akin to the way it is in x[sup]n[/sup] + y[sup]n[/sup] = z[sup]n[/sup].
My question, then, is can anyone cite some (non-trivial) theorems that are true for the first T cases (with T being “huge”) but are then violated for some V (with ‘V’ an even huger number)?
I don’t have an answer, but technically, theorem should refer to claims which have been proven generally, and so (aside from error) they can’t be violated by a sufficiently large counterexample.
Conjectures are more in line of your question–claims that are seemingly true because they hold for all easily checked cases, but not actually proven and thus vulnerable to counterexamples.
“Fermat’s Last Theorem” was thus a misnomer before Wiles proved it, but I guess maybe it got its name before people realized that Fermat couldn’t possibly have proved it with the tools available.
Depending on how you define “large”, Fermat numbers may fit the bill. They are numbers of the form 2^(2n) + 1.
It’s known that the first 5 Fermat numbers (up to 65537) are prime. The 6th Fermat number is 4294967297 and is not prime.
Fermat himself conjectured all Fermat numbers were prime but never proved it, and in fact Euler provided a factorization of the 6th Fermat number nearly 100 years later. Worse, all currently calculated larger Fermat numbers are composite. What is not known is whether or not they are all composite. It’s an open problem, as are several other questions about Fermat numbers.
Ian Stewart’s latest book, Visions of Infinity: The Great Mathematical Problems, discusses many problems where examples have been studied up to a very large number. Since they are problems there are no counterexamples, but I’m pretty sure he does mention ones where violations have been found.
I recommend the book, even though I stopped understanding even his simplified explanations of what was going on about halfway through.
The Fermat numbers get very big very quickly, but I’m not sure that they’re the most interesting case, given that there are only five Fermat primes before the first Fermat nonprime. Perhaps more interesting are the Fermat pseudoprimes base 2 (which despite being named after the same guy, are not particularly related).
Consider the statement “n>1 is a prime iff 2[sup]n-1[/sup] - 1 is divisible by n”. Go ahead, test it for a few numbers; it’ll probably work on every number you test. In fact, test it for any number up to 340, and it’ll work for each and every one of them, 339 successes in a row. But now test it for n = 341. 2[sup]340[/sup] - 1 is divisible by 341, and yet 341 = 11*31 and is thus not prime.
Ah, thank you, all. Skewes’ number may have been it (or not, see below). And thanks, Thudlow, for the link to that very similar thread.
Now, I will make no pretence of understanding exactly how Skewes’ number is obtained but will still ask: does not Skewes’ number simply put an upper bound on a number satisfying a particular relationship. If so, that is not quite what I was getting at. No, what I am wondering is along the lines of if there is/are conjecture(s) (;)) which hold for all n < T (where T is ‘very large’) but fail for some V >> T? (for ‘appropriate’ candidates for n, e.g. integers, n[sup]th[/sup]-prime, etc.)
Wikipedia states that 10[sup]14[/sup] is a lower bound for any x, π(x) > li(x). If I read Wikipedia correctly, not a single explicit example of x, π(x) > li(x) is known, but there must be some smaller than the Skewes’ numbers. IOW, the smallest such x has 10[sup]14[/sup] < x < 10[sup]316.3[/sup]
Thus, a “conjecture” that π(x) < li(x) would seem to satisfy your requirement, if you consider 100 trillion VERY large. This is in contrast to the construction for which the preposterously large Graham’s number is an upper bound. For that construction the lower bound is (at least until a few years ago) a mere 13. :smack:
That is Yesenin-Volpin’s conjecture. He believes that all sufficiently large numbers are meaningless (and totally rejects infinity).
The thing about Skewes’ number is that it is now known that not only does pi(x) exceed li(x) for some large x, but in fact the difference pi(x) - li(x) changes sign infinitely often. But not a single instance of sign change is known. I would assume that by now supercomputers could get up to 10^14, but I guess not.
If you want an explicit counterexample, you could try Pólya’s conjecture, which states that for all n, there are more numbers with an odd number of prime factors than an even number of prime factors. It fails to hold when n = 906,150,257.
