Answer by C7X for Important open problems that have already been reduced to a...
Is there always a prime in the interval $(x^3,(x+1)^3]$ for every natural number $x\geq 2$?Equivalently the interval may be changed to $[x^3,(x+1)^3]$. Assuming the Riemann hypothesis, this is provable...
View ArticleAnswer by Mark S for Important open problems that have already been reduced...
It is thought that it is more difficult to calculate the permanent of an $n\times n$ matrix than to calculate the determinant of the matrix. However, even for $4\times 4$ matrices this problem seems...
View ArticleAnswer by Mark S for Important open problems that have already been reduced...
At the turn of the 21st century, Catalan's conjecture that $8$ and $9$ are the only non-trivial consecutive powers was reduced to a finite but intractable problem. For example, it was known that to...
View ArticleAnswer by Gerhard Paseman for Important open problems that have already been...
Jacobsthal's function $g(n)$ for a positive integer $n$ gives the smallest number $g$ such that any interval of consecutive integers of length $g$ contains an integer coprime to $n$. I have been...
View ArticleAnswer by Per Alexandersson for Important open problems that have already...
In Conway's Game of Life, a lot of active problems are related to reducing the size of currently known types of starting states:It's well-known that there are diagonal (e.g. glider) and orthogonal...
View ArticleAnswer by Sam Hopkins for Important open problems that have already been...
As a result of Terry Tao's recent blog post the lonely runner conjecture for any particular value of $n$ has been reduced to a finite computation. Currently the LRC is verified only for $n \leq 7$.
View ArticleAnswer by Nick Gill for Important open problems that have already been...
You mention additive number theory in the question, so perhaps this isn't the type of example that you want. However my understanding is that the Three Primes Conjecture (every odd number $\geq 7$ is...
View ArticleAnswer by Jernej for Important open problems that have already been reduced...
I am not sure if this fits all the stated criterions but since it is a neat problem here it goes..Is there a 57-regular graph $X$ of order 3250 , girth 5 and diameter 2?$X$ is known as a Moore graphA...
View ArticleAnswer by Andrés E. Caicedo for Important open problems that have already...
This is an elaboration of a comment on Suvrit's answer. Ramsey numbers can be defined for (infinite) ordinals, just as in the finite case: $r(\alpha,\beta)$ is the least $\gamma$ such that for any...
View ArticleAnswer by Ian Agol for Important open problems that have already been reduced...
Thurston asked for the maximal number of non-hyperbolic Dehn fillings on a one-cusped hyperbolic 3-manifold, and conjectured that the maximum is 10 which is only achieved by the figure eight knot...
View ArticleAnswer by Joel David Hamkins for Important open problems that have already...
In principle, any mathematical question $\psi$ that is not independent of ZFC (or some standard stronger theory, such as ZFC+large cardinals) is reducible to the finite computational procedure: search...
View ArticleAnswer by David White for Important open problems that have already been...
Computing homotopy groups of spheres has been reduced in several different ways down to a finite but infeasible computation. This was discussed in another thread. John Klein's answer describes an...
View ArticleAnswer by Suvrit for Important open problems that have already been reduced...
Computing Ramsey numbers or even tighter bounds on them is perhaps a prototypical example that fits the bill.
View ArticleAnswer by Yoav Kallus for Important open problems that have already been...
Voronoi gave an algorithm to enumerate all perfect quadratic forms in $n$ variables and consequently to identify the densest lattice packing of spheres in $\mathbb{R}^n$.
View ArticleAnswer by joro for Important open problems that have already been reduced to...
Numerical evidence suggested $\pi(x)$ is always less than $\mathrm{li}(x)$.Littlewood proved that $\pi(x) - \mathrm{li}(x)$ changes sign infinitely often, but the smallest $x$ s.t. $\pi(x) >...
View ArticleAnswer by Tony Huynh for Important open problems that have already been...
One of the most important open questions in graph theory is Hadwiger's conjecture, which asserts that every graph with no $K_{t}$-minor is $(t-1)$-colourable. The cases $t=1,2$ are trivially trivial,...
View ArticleAnswer by Gerry Myerson for Important open problems that have already been...
Baker's work on linear forms in logarithms reduced great big families of diophantine equations to finite searches. In many cases, sharpening of Baker's results plus large amounts of cleverness have...
View ArticleImportant open problems that have already been reduced to a finite but...
Most open problems, when formalized, naturally involve quantification over infinite sets, thereby obviating the possibility, even in principle, of "just putting it on a computer."Some questions (e.g....
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