Unsolved Problems

Is $K \times [0,1]$ collapsible for every finite contractible 2-dimensional CW complex K?...

If k runners with distinct speeds run on a unit circle, will each runner be "lonely" (≥1/k away from others) at some time?...

Can the minimum size for sunflowers be bounded by an exponential (not super-exponential) function of k?...

For any finite union-closed family of sets, does some element appear in at least half the sets?...

What is the exact value of the Ramsey number R(5,5)?...

Is there a finite upper bound on multiplicities of entries >1 in Pascal's triangle?...

Do quasiperfect numbers exist?...

Do odd weird numbers exist?...

Are there infinitely many pairs of amicable numbers?...

Does iterating unsigned differences on prime sequence always yield 1 as first element?...

If Σᵢ aᵢᵏ = Σⱼ bⱼᵏ with m terms on left, n on right, is m+n ≥ k?...

For a polynomial $f(x) = ax^2 + bx + c$ with $a > 0$, $\gcd(a,b,c) = 1$, and discriminant $\Delta = b^2 - 4ac$ not a perfect square, the polynomial ta...

Let $\pi(n; a, b)$ be the number of primes $p \le n$ with $p \equiv a \pmod b$. For every $a$ and $b$ with $a \perp b$, are there infinitely many valu...

Let $\{a_i\}$ be any infinite sequence of integers for which $\sum 1/a_i$ is divergent. Does the sequence contain arbitrarily long arithmetic progress...

Are there arbitrarily long arithmetic progressions of consecutive primes? That is, for any positive integer $k$, do there exist $k$ consecutive primes...

Are there infinitely many Sophie Germain primes? A prime $p$ is called a Sophie Germain prime if $2p + 1$ is also prime....

Is it true that for infinitely many $n$, $d_n = p_{n+1} - p_n > c \ln n \ln \ln n \ln \ln \ln \ln n / (\ln \ln \ln n)^2$ for arbitrarily large constan...

Are there infinitely many twin primes? That is, are there infinitely many primes $p$ such that $p + 2$ is also prime?...

For any given pattern of primes with no congruence obstructions, are there infinitely many sets of consecutive primes with this pattern?...

Let $C(x)$ be the number of Carmichael numbers less than $x$. Does $(\ln C(x))/\ln x$ tend to 1 as $x$ tends to infinity?...