Mathematics Problem Archive

Chung/: A de Bruijn covering code of radius R is a binary string so that the set of words appearing as n consecutive symbols (with wrap-around) is a c...

Kolakoski: There is (essentially) a unique sequence over {1,2} which is its own run-length encoding. Is the density of 1's in this sequence 1/2? See t...

Alon: What is the threshold function n = f(k) for the event that a random permutation on n symbols contains all patterns on k symbols? Conjecture: f(k...

Tao: What is the probability that a random nXn matrix over Z_(p) has zero permanent as n goes to infinity? (Surely 1/p... as long as p is not 2.)...

Is the exponent of matrix multiplication 2? In other words, can two n b n matrices be multiplied in O(n^(2+)^(ľ)) steps? See this....

Let $A=(a_{ij})$ be an $n\times n$ matrix with nonnegative entries and total entry sum $n$. Define $$\phi(A)=\prod_{i=1}^n\sum_{j=1}^n a_{ij}+\prod_{j...

A superpermutation on $n$ symbols is a string containing every permutation of the $n$ symbols as a contiguous substring. Determine the minimum possibl...

For positive integers $N,q,a$, let $Q(N;q,a)$ be the number of perfect squares among $a,a+q,\ldots,a+(N-1)q$, and let $Q(N)=\max_{q,a\geq1}Q(N;q,a)$. ...

Let $W(r,k)$ be the least $N$ such that every coloring of $\{1,\ldots,N\}$ with $r$ colors contains a monochromatic arithmetic progression of length $...

For $n\in\mathbb{N}$, let $\Pi(n)$ be the product of its decimal digits, and let $\operatorname{Pm}(n)$ be the least positive integer such that $\Pi^{...

For a point set $P$ in $\mathbb{R}^d$, let $f_d(P)$ be the number of unit-distance point pairs: $$f_d(P) = \left| \{ (u,v) \mid u, v \in P, \, \|u-v\|...

For any (planar) triangle $T$, is there is a $3$-coloring of the (infinite) plane with no monochromatic copy of $T$? We imagine congruent copies of $T...