# How shall we handle big numbers / computer searching problems?

There are certain questions that ask about finding an object with a certain property, usually some large number. Now, to me, it is on the edge of what this site is about, as the question is not really about understanding the math, but rather asking for help with computation (although sometimes an answer can be given by purely mathematical reasoning). Just to give a few examples:

Do we have some recommended approach to these types of questions? I was thinking if perhaps there is a better platform for these elsewhere. However, they do not even seem to fit sites like Art of the Problem Solving (AoPS) forums. This is a reason I personally do not downvote these, as I do not know where else these should be asked. However, at least by the downvotes it seems lots of users have an opposite opinion.

If these types of questions are worth keeping on this site, should we at least have some tags to recognize these? Closest thing I found related is [big-numbers] and [examples-counterexamples], not sure if either of them fits.

• Are you suggesting users are downvoting because they think computer searches don't belong on m.se? I can see other reasons. The 4th question is utterly lacking in motivation. The 3rd shows an ignorance of a well-known problem (though it would take a mean user to downvote for that reason). The 2nd misuses the term cube full (although again it would take a mean user etc., etc.). The 1st invited a downvote over the use of the term probable prime. – Gerry Myerson Dec 6 '19 at 9:54
• @GerryMyerson Neither of those seem to me enough reason to downvote, except the lack of motivation which seem to be common across these questions (although what motivation could one expect since these are just about a searching of number with particular property...). So yes I am suggesting people are downvoting these questions as they do not fit, but I guess it should be up to them to explain... – Sil Dec 6 '19 at 10:02
• Maybe that could be output of this (meta) question, what tags, and motivation/context is to be expected on these questions? – Sil Dec 6 '19 at 10:09
• The user posting the 4th question could have mentioned that $40$-factorial was special because it's so close to a power of $1,000$ which leaves very little room for three prime factors with the same number of digits. – Gerry Myerson Dec 6 '19 at 10:14
• There is much math in programming... – user645636 Dec 6 '19 at 15:29
• Programming for the purpose of computation is math. Doing a computation doesn't cease being math because you can't do it with a pencil and paper. – Matt Samuel Dec 6 '19 at 20:31
• @MattSamuel Likely you mean "purpose of mathematical computation". But even that needn't be considered mathematics by mathematicians, e.g. numerology. – Bill Dubuque Dec 7 '19 at 4:10
• @TheGreatDuck I disagree; I don't think that "I'm curious" is a good enough reason. Questions on MSE should be of use to someone other than just the asker. If the only reason that an asker has for posting a question is that they are curious---that is, if they can't make a cogent argument about why the question might be more broadly appealing---then they should not be asking that question on this platform. – Xander Henderson Dec 7 '19 at 19:22
• "they couldn't have mentioned that because that is your thought and not their thought. If they didn't have that thought, then they shouldn't put it in their question." @The did you overlook the comment Peter posted, "I also noticed that the fact that $40!$ is "near" $10^{48}$ makes the search longer." – Gerry Myerson Dec 7 '19 at 21:31
• @TheGreatDuck As soon as you show me a hard rule or policy which indicates that mere curiosity is sufficient. Until then, we are both merely expressing opinions. – Xander Henderson Dec 8 '19 at 0:47
• That being said: "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." (emphasis mine; note that questions should be relevant to the asker and the community). – Xander Henderson Dec 8 '19 at 0:48
• @TheGreatDuck Your response to me appears to be a complete non sequitur. You stated that if a user says "I'm curious about this...", that should be sufficient motivation. I disagreed. I stated my opinion that I think that the motivation for a question needs to be more than the curiosity of the asker---a question have broader appeal. This has nothing to do with a question being homework or engineering or whatever other criteria you invented. It is about whether or not the question is interesting anyone other than the asker. – Xander Henderson Dec 8 '19 at 1:36
• coding, etc. rely on such sanity checks @The – user645636 Dec 23 '19 at 16:30
• The most upvoted question I've ever asked was Proof that $\left(\pi^\pi\right)^{\pi^\pi}$ is a noninteger. I think it was a good question, and Alexander Walker's answer to it, which I thought was superb, was well-received. – MJD Dec 26 '19 at 18:42

We should keep these problems, because you never know when you have one.

Some days ago, user Nikita asked the question Is this contrived sequence eventually periodic?. Innocently enough, they explained that they had built quite a weird sequence $$A(n)$$, which seemed to have a period, as all residue classes mod $$6$$ except for one gave the exact same thing. I then showed that all residue classes mod $$60$$ except for one, also gave the exact same thing. All was looking ok.

Long story short, in the comment section to this answer by WhatsUp, we proved that, if the sequence was periodic, its period was at least $$5354228880$$. There's no way that an answer to this problem can escape computational machinery at this point.

The lesson? In math, you never know if a problem will have a beautiful and ingenious argument, or if after checking the first $$4.1\cdot10^{18}$$ cases, a counterexample will appear. This is also true for questions in this site. We shouldn't do any special maneuver just because we haven't been able to discard either option.

• @postmortes Those are two very different concepts. Being a hoarder means keeping something “because you might need it”. I’m suggesting keeping something “because you might not know if it has a certain generally undesirable characteristic”. – URL Dec 9 '19 at 8:39
• @postmortes your analogy fails because of the inherent difference between a cluttered house and a computer database with an algorithm that automatically floats popular content to the top. With the consequence that that what is hoarded in this site is actually a highly organised library with a central atrium for the popular material but deep and extensive vaults, thoroughly indexed, and a large community of committed volunteer librarians. – samerivertwice Dec 10 '19 at 10:11
• @postmortes my proposition is that the in-built upvote, downvote system, detailed search functionality, user reputation, duplicate question system and topic tagging framework already do combine to make this platform a powerful and effective way of cataloging this kind of info. The stuff you don't want to see right now naturally bubbles down out of view but is there if you search for it. – samerivertwice Dec 10 '19 at 10:58
• As someone who spent roughly 7 years helping out in school libraries, you give volunteers too much respect. Also your hidden shelf analogy fails, it's often most requested, it has textbooks and dvd's . – user645636 Dec 10 '19 at 11:23
• oh and the strong version of Goldbach is equivalent to that at least one of the partitions for even values, into 3 primes is 2. – user645636 Dec 20 '19 at 2:05
• @postmortes If a hoarder owner a 2 acre sized mansion they might have enough furniture to make a single house “hoarded”, but they are not a hoarder. They have a reasonable quantity. It is the same with this site. Unless stack exchange says “raise the alarm we are out of hard drive space” there is no rational reason to delete posts without a decent argument for it. The hoarder argument of “you can’t keep it or else you’re being a hoarder” only applies when what you own is becoming excessive. – user64742 Dec 23 '19 at 19:41

Computer searches are a part of modern mathematics. It is natural for people to ask about them. Much of math research is about finding patterns and making conjectures.

In addition, some of the questions in the OP could be solved without computer searches, for example Consecutive sequences of not cube-free numbers which could be solved nicely with the Chinese Remainder Theorem. The question asker doesn't know this a priori so is justified in asking.

As a side note, https://math.stackexchange.com/tour also explicitly encourages asking about software mathematicians use.

• PARI GP, Mathematica ( has own stack exchange), Wolfram alpha, Java applet (at one point by a mersenneforum user). basically any programming language that can do the job ( GIMPS uses ASM and C a lot at last check). – user645636 Dec 16 '19 at 11:34

To weigh in on the tag, the tag description is:

For questions relating to the computation, estimation and properties of extremely large finite quantities that are not usually used in mainstream mathematics. This is not for questions that just have large numbers; the fact that a number is very large has to affect the question.

As the last sentence states, this is not simply meant for questions that have large numbers in them. Rather, the fact that the number is large must affect the question at hand. For example, the currently most recent question asks for a way to compute the last few digits of Graham's number. Usually, the computation of $$a^b\bmod c$$ can be easily done using known algorithms with a computer. However, the sheer magnitude of Graham's number simply makes this not possible. That is to say, this tag is usually meant for questions where the numbers themselves are so large that directly working with them, either by hand or by computer, is impossible. These are certainly not your average "I need help with a quick computation" question, and usually do indeed require things such as inductive proofs.

• Okay, so perhaps we need a new tag? Although judging by number of upvotes/downvotes in this meta seems this is a bit controversial (although unfortunately none of the downvoters posted an answer to clarify). – Sil Dec 17 '19 at 3:17
• power towers only last ceiling of the log of the modulus high at most in modular arithmetic. – user645636 Dec 17 '19 at 21:15