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I am fascinated by mathematical conjectures, especially those that are believed to be false. Now I wonder, are there some mathematical conjectures for which the numerical search would be promising? What would be conditions for such conjectures? For example, there shouldn't have been extensive search already, and it should be (widely?) believed to be false.

For instance, I think the Riemann hypothesis is a bad example (much investigated and not believed to be wrong), but Firoozbakht's conjecture might be a good canditate (believed to be wrong, and not so famous).

Are there conjectures which fulfill those, or other conditions of that kind?

Edit: A very good example has just been published, namely the Boolean Pythagorean triples problem, where the researchers have used a computer to check roughly one trillion different possible solutions and found a negative result (see more in the Nature News&Views article).

Edit2: Another very nice example has just been presented at Numberphile, asking whether or not numbers can be written as the sum of three cubed numbers.

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Mario Krenn
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  • you could focus of the conjectures of the form "the program $P()$ doesn't halt" with $P$ a particular program running on some Turing machine. the Riemann hypothesis and most of the conjectures on the prime numbers are of this kind. however, if proving those conjectures are false can be simply done by running the program, proving those are true cannot be achieved in general by numerical search (it doesn't reduce to "the program $Q$ halts") – reuns Apr 29 '16 at 17:37
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    The Riemann hypothesis is exactly a conjecture which is not promising, because there has been already extensiv computer search, and it is widely believed to be true (or at least, no counterexample might appear). – Mario Krenn Apr 29 '16 at 18:31
  • the point of my comment was that : if a conjecture reduces to "the program $P()$ does/doesn't halt" then computer search is promising for proving that it halts, but not so much for proving that it doesn't halt. https://en.wikipedia.org/wiki/Halting_problem – reuns Apr 29 '16 at 19:08
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    I clarified my question. Sorry, but I still think that your comment does not go into the direction that I'm looking for. Obviously each conjecture is actually a question of whether there is a counter-example (if a program searches for it forever or not) - that is simple and not what i'm asking. I'm asking for conjectures that are more likely to be shown by computers that they are wrong. So your example (Riemann hypothesis) is a very bad example (because it has been investigated a lot numerically, and is believed to be not wrong), but for instance Firoozbakht's conjecture might be a good one. – Mario Krenn Apr 29 '16 at 19:14
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    I wonder why this question has received a downvote and no upvotes other than my own. I guess it might be because people think that there cannot be such problems because otherwise someone would already have carried out the numerical search to a point where it would no longer be promising. However, this argument seems circular. The reach of numerical searches is constantly expanding, and if no-one ever thought about which numerical searches are currently promising, no-one would be extending them. – joriki Apr 29 '16 at 20:42
  • Thank you joriki, i'm glad that you understood (and expanded) my intention of the question. Do you have an idea how to improve it such that more people might find it interesting to think about it? And do you have an idea about the conditions for such conjectures, or even have some fitting examples? – Mario Krenn Apr 29 '16 at 23:54
  • Have you seen anything about the collatz conjecture? – user328442 May 12 '16 at 15:49
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    Firoozbakht's conjecture has been verified for primes less than $4\times 10^{18}$ and even that was not purely computational - and, far as I can tell, this calculation was only limited by calculations about prime gaps. I think it would be difficult to extend a numerical search. – Milo Brandt May 12 '16 at 15:59
  • Thank you Milo, that's a very interesting point. So my example was not very good, do you know of other examples which might be better? – Mario Krenn May 13 '16 at 07:50
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    Why is it necessary that its believed to be false? you can just take the converse of one believed to be true to get a conjecture believed to be false –  May 18 '16 at 12:54
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    There are some sites that host list of numbers believed to be prime, and probabilities of them beeing prime, and colloective computer searches are ongoing. How about, white has a forced win in chess? Or some other game. –  May 18 '16 at 13:00
  • Is Firoozbakht's conjecture generally believed to be false? There are of course non-rigorous arguments against it but I've never heard anyone claim it is "believed to be false." – M10687 May 18 '16 at 19:56
  • @Loffen Interesting what you say about these sites, do you have some links? "Believed to be false" because "It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of Granville and Pintz" (from wikipedia). The chess-example is a very good one, do you know whether there is a distributed-computer-search for that? – Mario Krenn May 19 '16 at 14:14

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(Posting this as an answer, in part because the comments thread is already too long.) Here is a relatively new open question (2016) where a numerical search for counterexamples seems promising:

Let $q$ and $r$ be fixed coprime positive integers, $$ 1 \le r < q, \qquad \gcd(q,r)=1. $$ Suppose that two prime numbers $p$ and $p'$, with $p<p'$, satisfy $$ p \equiv p' \equiv r \ ({\rm mod}\ q), \tag{1} $$ and no other primes between $p$ and $p'$ satisfy $(1)$. Then we have the following

Naive generalization of Cramer's conjecture to primes in a residue class: $$ p'-p ~<~ \varphi(q) \log^2 p'. \tag{2} $$

(PrimePuzzles Conjecture 77, A. Kourbatov, 2016). See arXiv:1610.03340, "On the distribution of maximal gaps between primes in residue classes" for further details, including the motivation for the $\varphi(q)$ constant. As usual, here $\varphi(n)$ denotes Euler's totient function.

Note: The logarithm in $(2)$ is taken of the prime $p'$ at the larger end of the "gap" $[p,p']$. (If instead we take the log of $p$, the smaller end of the "gap", then counterexamples are easy to find.)

Alex
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  • A counterexample is now known: $$q=1605, \ r=341, \ p=3415781, \ p'=3624431.$$ (Conjecture $(2)$ is very likely to be true "almost always".) – Alex Sep 20 '17 at 11:06