An often repeated fact of mathematics is that just because a conjecture seems to hold for a very large amount of values, that doesn't mean that it's true. And to demonstrate this, there are examples of such conjectures which have been made, and seemed to hold for really large amounts of values, but ended up being false conjectures after all.
The most typical example of this is the Pólya conjecture: If we partition all natural numbers up to some
n into those with an odd number of prime factors and those with an even number of them, it seems to be that there are always more of them in the odd side than the even side. This was conjectured to be so. But it turned out to be false. And the smallest
n for which it's false is 906150257.
The
offset logarithmic integral function Li(x) is a function that approximates the prime-counting function π(x). (In mathematical notation, π(x) ∼ Li(x).) It was noticed that it seems that Li(x) always gives larger values than the actual prime-counting function, and this is true for a staggeringly large amount of values. Thus it was conjectured that it's always so. However, once again, this is not true. It has been proven that for an x somewhere between 10
17 and 10
316 Li(x) becomes smaller than π(x).
The smallest possible counter-example to the
Euler's conjecture for k=4 is: 95800
4 + 217519
4 + 414560
4 = 422481
4
I was wondering if there are any other seriously proposed conjectures which seem to hold for very large amounts of values, but which nevertheless have a counter-example (which is very large).
