> The Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10^361.[3]
> An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960;[4] the smallest counterexample is n = 906,150,257, found by Minoru Tanaka in 1980.[5]
EDIT: rephrased because despite having a maths degree I can't count.
That has not born fruit so far.
In hindsight I could have had the same career path with a straight CS degree and much less stress during my college years. No regrets though, math really is fun! Even more so when the classroom setting is removed.
At school, we had tests to show what we were good at. The guy who came to comment on our tests told me that I should definitely go for something like literature or history.
And here I am, an engineer with an extra PhD in physics who dreams about meeting these idiots and explaining to them that what they do is revolting. They are probably dead by now though (it was in the 80's)
Had to look it up.
Reminds me of a story from <https://www.ams.org/notices/200410/fea-grothendieck-part2.pd...>. Quoting it below:
One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”
It's like they say: the only numbers a mathematician needs are 0, 1, and 2 (and just because it's not 1).
(And there's always unary.)
For instance, under this notation, the dual of L^p is just L^(1-p). And Littlewood’s interpolation inequality is a lot easier to remember, since the exponents used come directly from the coefficients in the convex combination:
If r = ap + bq, where a and b are nonnegative and sum to 1, then |f|_r <= (|f|_p)^a (|f|_q)^b
And going off on a tangent: in thermodynamics, we should measure coldness (coldness ~ 1 / temperature). It makes all the math come out nicer.
See https://en.wikipedia.org/wiki/Coldness
Coldness handles 'negative temperatures' much better. As Wikipedia puts it:
> Though completely equivalent in conceptual content to temperature, β [= coldness] is generally considered a more fundamental quantity than temperature owing to the phenomenon of negative temperature, in which β is continuous as it crosses zero whereas T has a singularity.[7]
1.845 × 10361 = 19116.045
You would probably enjoy Tom Körner's book 'The Pleasures of Counting'. See eg https://maa.org/press/maa-reviews/the-pleasures-of-counting
[1] https://en.wikipedia.org/wiki/Collatz_conjecture?useskin=vec...
The prime number theorem states that the prime counting function $\pi(x)$ is well approximated by the integral $\int_0^x \frac{dt} {\log t},$ which is a function that's become named $\mathrm{li}(x).$ Littlewood proved that the sign of $\pi(x) - \li(x)$ changes infinitely often, but his proof didn't produce a specific value where such a sign change occurs.
In 1933, one of Littlewood's students showed that, assuming the Riemann hypothesis, at least one sign change occurs below the number $e^e^e^79$, which is approximately $10^10^10^34.$ In 1955, he was able to show unconditionally that such a sign change occurs below the number $e^e^e^e^7.705$, which is approximately $10^10^10^964$.
Both of those numbers absolutely dwarf the estimated number of elementary particles in the observable universe, so it's utterly impossible to either compute or write them down, even if you used the entire universe as your computer or your writing surface.
https://en.wikipedia.org/wiki/Skewes%27s_number
That upper bound has since been lowered in 1999 to 1.38922 * 10^316 here: https://www.ams.org/journals/mcom/2000-69-231/S0025-5718-99-...
I imagine it there might have been a bit more progress in the last 25 years, but I have no idea if it's possible to actually write down all the digits of the current best upper bound.
I'm probably missing something that should be obvious, but why wouldn't it be possible to write down 317 digits?
There might be a problem with actually calculating the 317 digits though. ¯\\\_(ツ)\_/¯
The wiki article on Skewes number also notes that the bound has been lowered to 1.397162×10^316 in 2011.
Which seems a bit paradoxical. If you can prove that the Collatz conjecture is undecidable, you would also prove that it has no counterexamples, and thus that it is true. Which would make it decidable -- contradiction. So this seems to prove that if the Collatz conjecture is undecidable, this fact is itself also undecidable.
> if you generalise the coefficients
The result appears to be https://gwern.net/doc/cs/computable/1972-conway.pdf if you want to read in detail
That is the case for something like Goldbach's Conjecture, which says that every even number > 2 is the sum of two primes. If it's false, then there is a counterexample, and it is easy to prove whether or not a given number is a counterexample (just loop over all pairs of smaller primes).
But that is not the case for the Collatz Conjecture. A Collatz counterexample could be a number whose orbit loops back around. That would be a provable counterexample. Another kind of Collatz counterexample would be a number whose orbit never terminates or repeats, it just keeps going forever. If such an infinite sequence existed, it might not be possible to prove that it's infinite. And if it isn't provable, then the conjecture would both undecidable and false.
> the smallest counterexample is n = 906,150,257, found by Minoru Tanaka in 1980.[5]
why on earth would you ever think that
I think that answers your question?
The point was more that any length of time greater than a few thousand years is as good as infinite for a human, so (for, you know, everyday purposes) we might as well assume that the sun will keep on rising forever.
It seems that for such a simple problem, involving basic arithmetic and small numbers, 1, 2, and 3, there should be a number N where if you try all examples less than N, you have sufficient "resolution" to reveal all patterns between the numbers. Maybe we could somehow say "if a counter-example exists, it must be smaller than N". I don't know what kind of math would allow us to formalize the possible patterns and what N is, I don't think it's been discovered yet. Maybe such math doesn't exist, but certainly we haven't discovered all mathematical tools for solving such problems, so maybe it does exist.
This kind of assumes that “A simple problem, involving basic arithmetic and small numbers, 1, 2, and 3” can be encoded into some bounded-state Turing machine.
it would be trivial to artificially construct a series that equals 1 until n=10^10 at which point it equals pi/785498. if these series do exist, then - without additional information - why would you ever think that one you're studying is not one of them?
even from a metaepistemological standpoint, had you never come across the notion that, for example, Fermat's Last Theorem could not have been proven simply by showing that 10^10/infinity of the possible outcomes had been shown to follow it?
>It seems that for such a simple problem, involving basic arithmetic and small numbers, 1, 2, and 3, there should be a number N where if you try all examples less than N, you have sufficient "resolution" to reveal all patterns between the numbers
does it? why?