Re:How many real numbers exist? New proof moves closer to an...
by sillysaurusx on Thursday July 15, 2021 @01:01PM
(#27846939)
https://math.stackexchange.com/questions/514/conjectures-tha... was excellent. Thank you.
> 𝑛^17+9 and (𝑛+1)^17+9 are relatively prime
> The first counterexample is 𝑛=8424432925592889329288197322308900672459420460792433
I think this helped me appreciate the difficulty of being satisfied with the Collatz conjecture.
There are two reasons: firstly, because mathematics is not an empirical discipline (well, unless you're a number theorist...), so it is possible to be certain of mathematical truth, unlike the inherent uncertainty of physical truth; secondly, because every finite bound on the natural numbers may as well be 0 when compared to the numbers that remain.
There is simply no way to take any empirical measurements of the natural numbers (or the reals) that would let you estimate anything "for all numbers", since the proportion of numbers you failed to sample is infinite.
You may be interested in reading the answers to this question: https://math.stackexchange.com/questions/514/conjectures-tha... which describes some problems which seemed true "up to some large number", but later turned out to be false.