Largest counter example to a mathematical conjecture

The most recent Abstruse Goose had me wondering whether there’s ever been a case of a mathematical conjecture in which a the first counter example has been very very large.

For example, something like Goldbach’s Conjecture has been machine checked up to 10[sup]18[/sup] without finding a counter example. Is there really any chance we’re going to find a counter example if we keep on looking?

Borsuk’s Conjecture is not known to fail until n=298.

Well, Lehman’s counterexample to Polya’s conjecture is 906180359, which is pretty large.

In some sense, the classification of finite simple groups fits this question. Maybe if we conjectured that there were infinitely many ‘sporadic’ groups and therefore the finite simple groups are not effectively classifiable? In any case, the biggest one has order

246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
= 808017424794512875886459904961710757005754368000000000

http://en.wikipedia.org/wiki/List_of_finite_simple_groups

Allow me to boldly conjecture that all numbers are less than 10[sup]402[/sup]. (I haven’t yet been able to prove this, but I’ve machine-checked it up through quite a large range; if there are any counterexamples, they’d have to be very large indeed)

Scrooge McDuck’s wealth has been authoritatively given as five billion quintiplitilion unptuplatillion multuplatillion impossibidillion fantasticatrillion dollars. I believe you’ll find that this is a larger number, despite a recent error in calculation by Fortune magazine.

Sorry, but it is well established that Scrooge McDuck’s holdings are almost entirely limited to the planet Earth, and therefore quite modest in total volume. (His claim to “own” the Andromeda Galaxy is still working its way through the courts).

In total volume, maybe, but what about hypervolume? His Money Bin is clearly six-dimensional, seeing as it has a capacity of three cubic acres.

Can one boldly conjecture, or must one either boldy conject or perhaps choose to issue a bold conjecture?

Of course.

Yes, but there is a difference between a proposition that fails only for very large numbers and a proposition whose truth or falsehood has been settled by the discovery of a very large counterexample.

10[sup]402[/sup] + 1

Ha-HAH!

I conjecture that there are no numbers greater than 10[sup]402[/sup] + 1.

Is there?

Do you have a counterexample?

“conject”? Anyway, dictionaries back up my experience of “conjecture” as (in addition to its use as a noun) the verb of choice here.

What do you mean by this as relates to the conjecture I put forth?

Skewes number (see http://en.wikipedia.org/wiki/Skewes’_number) is something like 10^{10^{10^{34}}} was the original number that contradicted the conjecture that li(x) < pi(x). Let me explain. pi(x) is the number of primes less than x and is asymptotic to li(x) (that is their ratio tends to 1 as x goes to infinity). I will attempt to explain li(x) below. But first, it had been calculated that for all “ordinary” numbers, that is numbers you could actually calculate with, li(x) > pi(x) and it was quite natural to suppose that inequality always held. Well it doesn’t and Skewes proved that there was at least change of sign of li(x) - pi(x) before what got to be called Skewes number. Better bounds are now known, but no actual counter-example has been found. As for li, the logarithmic integral of x, it is the integral from 2 to x of 1/ln(x).

I’ve heard it said that Skewes number is the largest number that ever arose in a serious mathematical context.

True at the time, but it stopped being true many years ago.

Graham’s number is far, far larger. In fact, with Graham’s number, you don’t even really need the “serious mathematical context” qualifier, since it’s so big that non-mathematicians can’t even conceive of the mental gymnastics that would be needed to come up with a bigger number.

Is there any such thing as a number that is finite but “inexpressibly huge”? That is, it is so large that all human attempts at notation of it are hopelessly inadequate?

Interestingly, this is the concept of an “unenumerable number” which in ancient and medieval India was investigated by Jain mathematicians. They considered it an intermediate category of number between finite and infinite.

In terms of modern mathematics, though, I don’t see why it would be theoretically impossible to come up with a notation to express any finite number. Maybe it could be shown that there are some numbers that are so big that no notation could express them except by using so many terms that they would be realistically impossible to state in any reasonable time period. But surely if a finite (rational) number exists, it is theoretically capable of being numerically expressed in some way or other. Surely?

In practical terms, of course, the expressibility of a large number depends on the flexibility of your number system. The ancient mathematician Archimedes wrote a book to demonstrate that a notation could be constructed that could express the number of grains of sand it would take to fill up the universe (of course, he thought the universe was considerably smaller than we now do). He had to put in some work to be able to express numbers going up even as far as ((10^8)^(10^8))^(10^8), which is fairly sizable but insignificant in comparison with something like Graham’s number.