The seventh most popular easily understood unsolved problem on MathOverflow

Post by BHSPitMonkey » Sat, Feb 17, 2024, 7:38 AM UTC

Paul-Craft wrote: post #69

Buttons840 wrote: post #8

One of the comments was "the Collatz conjecture feels like we're missing a branch of mathematics", or something to that effect. I followed that rabbit hole a bit and found the Plya conjecture[0], which was disproven when a counter example was found at approximately 10^361. Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist. If only it…

One of my favorite mathematical tangents involves a theorem of Littlewood. 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 ch…

Post by Paul-Craft » Sat, Feb 17, 2024, 8:07 AM UTC

BHSPitMonkey wrote: post #71

Paul-Craft wrote: post #69

Earlier quoted context omitted.

One of my favorite mathematical tangents involves a theorem of Littlewood. 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 ch…

> That upper bound has since been lowered in 1999 to 1.38922 * 10^316 here: ... > > 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?

Post by cubefox » Sat, Feb 17, 2024, 9:45 AM UTC

nickdrozd wrote: post #43

cubefox wrote: post #31

Earlier quoted context omitted.

If it is "undecidable" this means there is no counterexample to the Collatz conjecture, since any counterexample would disprove it. But the Collatz conjecture does exactly state that there are no counterexamples. Which means: If it is undecidable, it is true. 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 i…

> If it is undecidable, it is true. 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 numbe…

Post by renonce » Sat, Feb 17, 2024, 10:40 AM UTC

zeroonetwothree wrote: post #45

tzs wrote: post #38

Here's a fairly easily understood problem, which if you could solve it would make you famous in the mathematical world and win you a million dollar prize: For a positive integer n: Let H(n) = 1 + 1/2 + 1/3 + ... + 1/n Let D(n) = the sum of the positive integers that divide n. E.g., D(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. Prove or disprove that for any positive integer n > 1: D(n) That easy to understand problem turns ou…

What a weird coincidence, I was just looking at this paper today for a different reason.

Post by EVa5I7bHFq9mnYK » Sat, Feb 17, 2024, 12:17 PM UTC