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Comment by ColinWright | Hacker News Reader

(a) The test is the wrong way round for the Collatz Conjecture;

(b) The Collatz Conjecture isn't one of the Millennium problems;

(d) It seems reasonable to say "If this routine exits then the value will be "true";

(e) None of what I say here is deep or new, so when people get excited about this sort of thing it makes me wonder what I've missed.

So ... what have I missed?

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(a) part of the joke

(b) the Millennium problems are not the only million-dollar prizes in mathematics

(d) part of the joke (and part of the C++ standard)

(b) There are other prizes, but I'm not aware of any others worth a million dollars. Do you have any pointers? A quick interwebs search didn't turn up anything, but they might just be being swamped by the results with the Millennium Problems.

> A quick interwebs search didn't turn up anything [...]

Searching 'collatz conjecture prize' quite literally puts a prize of a bout 120MM Yen (~ $1MM) as the top result.

Hah! I didn't search specifically for that, I was searching for bounties in general.

Thanks.

There is also a Beal Conjecture[0] worth a million

[0] https://en.wikipedia.org/wiki/Beal_conjecture

My initial search for bounties didn't turn up that one, either. I wonder if my connection(s) with Erdos have polluted my search history, so it was showing me mostly those, and not showing me this.

<shrug>

Thanks for finding this and letting me know, I've learned something.

The joke.

Savage. I dont often laugh loud reading HN. Today I did. Thanks.

I've only seen Veretasium YouTube video on this a few days ago, and I'm not a mathematician. Is the joke that LLVM has solved the mystery by optimizing out what it decided is an infinite loop?

Yes.

The C++ standard allows compilers to assume that loops terminate. Since there is only one way to exit the loop (`return true;`) the compiler can assume that this line will eventually be reached. Since the rest of the loop has no side effects, the compiler can optimize it out.

So the compiler doesn't actually need to figure out if the loop is infinite or not.

> The C++ standard allows compilers to assume that loops terminate.

Technically, it doesn't, but it requires programs to make forward progress, which is defined as either terminating, calling I/O functions, accessing a volatile variable or performing an atomic or synchronization operation. Infinite loops that'll eventually do one of these are perfectly valid C++ and don't have to terminate.

It optimizes "waste an unpredictable time doing arithmetic without side effects, then return 1" to "return 1 without looking at the input". The relation between that machine arithmetic and the Collatz conjecture is irrelevant.

I don't think you're responding to the right comment?

LLVM optimizes out the infinite loop, because the C++ standard defines that all programs must be able to make forward progress and so removing this loop does not alter the result of the program. Thus, collatz() as written in the tweet will always return true when optimized by LLVM. This says nothing about the conjecture being true or not, it is solely an artifact of the C++ standard being written a certain way.

But also a layer deeper, collatz() takes a uint128 as input and the Collatz conjecture has already been proven to be true for all numbers that would fit into a uint128. So LLVM arrives to the correct answer (`return true`) because of the wrong reason, mathematically speaking.

This doesn't exactly follow the Collatz conjecture because multiplication by 3 will be truncated to a 128-bit unsigned int. It is possible that this truncation causes a loop. Even though the Collatz conjecture has been tested for all numbers in this range, it has probably not been tested with this overflow effect.

Edit: removing my comment, I misunderstood.

Not so, since the fact that it is in the range does not mean it will steadily go downwards. It might have numbers much bigger than the maximum value for a uint128 somewhere in its Collatz sequence and it is possible (though unlikely) that one of those numbers modulo the maximum value for a uint128 is the original number again. (That is, the modulo operation might introduce loops that are not present in the normal Collatz sequence because mathematical integers don't overflow)

> The test is the wrong way round

Specifically, Collatz says "if x is even then x /= 2" but the code checks "x is odd".

> (b) The Collatz Conjecture isn't one of the Millennium problems;

It's not but there are more bounties out there than just the Millenium list. For the Collatz Conjecture specifically there's a 120M Japanese Yen prize which at this moment is a bit more than 1 million USD.

https://www.prnewswire.com/news-releases/bakuage-offers-priz...

(Also commented elsewhere) ... I didn't search specifically for that, I was searching for bounties in general.

Thanks.

> (c) It's using bounded numbers in the range where the Collatz Conjecture is known to be true;

The input is an unsigned __int128, and I believe Collatz has only been verified up to 2^68.

Not to mention that the answer by itself is pretty much worthless, but Mathematicians are rather interested in a proof of the conjecture (or its negation).

I get that it is supposed to be a joke, but it kind of falls flat because it misses the point entirely.

You have no sense of humor or curiosity about compiler optimization details that led to this, I guess.

For instance not all languages that target LLVM will get this result. That is interesting.

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