Mathematics Problem Archive

Let $\zeta_p:\mathbb{Z}_p\to\mathbb{Q}_p$ be the $p$-adic zeta function. Is $\zeta_p(k)\ne0$ for every even integer $k$?...

Given an integer $n$, determine whether there are rational numbers $x,y,z$ satisfying $x^2+y^2=z^2$ and $xy=2n$; equivalently, determine whether $n$ i...

Is every integer $n\equiv5,6,$ or $7\pmod 8$ a congruent number?...

For $E_n:y^2=x^3-n^2x$, is $n$ a congruent number if and only if $L(E_n,1)=0$?...

Given an elliptic curve $E:y^2=x^3+Ax+B$ over $\mathbb{Q}$, determine whether $E$ has infinitely many rational points....

For an elliptic curve $E/\mathbb{Q}$, let $N_p$ be its number of solutions modulo $p$ plus one and put $f(X)=\sum_{p\le X}\log(N_p/p)$. Is $f(X)$ boun...

For an elliptic curve $E/\mathbb{Q}$ of rank $r$, does $f(X)=\sum_{p\le X}\log(N_p/p)$ grow asymptotically like $r\log\log X$?...

For a compact convex domain $\Omega$, the values of $F_\Omega$ at the vertices of its corner locus $C_\Omega$ give complete coordinates. How are these...

Find a higher-dimensional analogue of the paper's cropping and summation argument; in dimension three the expected sum ranges over quadruples $v_1,v_2...

For $Z(s)=\sum f(a,b,c,d)^s$, which is known to converge for real $s>1/2$, extend $Z$ to complex values of $s$....

Give another proof of the paper's identities (Ж) and (ж) using the methods for identity (1). Can $f(a,b,c,d)$ be interpreted as a residue at $(a+b)+(c...

Can the function $f$ on $SL(2,\mathbb{Z})$ be extended naturally to $\mathbb{C}/SL(2,\mathbb{Z})$? Can analogous series be constructed for other latti...

For every odd prime $p$ and $k\ge1$, is $s(p^k)=k$? For $k\ge2$, is $d(p^k)=p^{k-1}d(p)$?...

For every $k\ge1$, is $s(2^k)=u_k$? Is $d(2^k)=2^k$ for $k\ne2$, with $d(4)=2$?...

Is Arnold's sequence $(u_k)_{k\ge1}$ the $f$-transform of the quadruple $(2,4,4,4)$?...

Call an infinite connected graph $G$ of uniformly bounded degree an infinite expander if there is a constant $c>0$ such that, for every vertex set $S$...

Does there exist a graph $G$ and a map $f:V(G)\to\mathbb{R}^2$ such that $|\|f(x)-f(y)\|_2-d_G(x,y)|<C$ for every $x,y\in V(G)$ and some constant $C<\...

Must every vertex-transitive graph of exponential growth contain an infinite subgraph $H$ with positive Cheeger constant $h(H)>0$? Can $H$ always be c...

Must every graph with uniform exponential volume growth contain an infinite subgraph, possibly a tree, having positive Cheeger constant?...

Prove that every bounded-degree transient hyperbolic graph contains a transient subtree....

Does there exist a vertex-transitive graph whose multiplicative rough-isometry constants to its $k$-net graphs tend to $1$ as $k\to\infty$?...

Does there exist a graph for which repeatedly passing to $k$-net graphs, at appropriately chosen scales, produces strictly smaller large-scale graph m...

Let $G$ be vertex-transitive and let $G_k$ be a $k$-net graph of $G$. Must $h(G_k)\ge h(G)$ whenever $G_k$ is not a single vertex?...

There is a positive function $f(h,d,k)$ such that every graph $G$ with $h(G)>h>0$ and maximum degree less than $d$ has $h(G_k)>f(h,d,k)$ for each $k$-...

Must every Cayley graph of superlinear growth contain, up to rough isometric embedding, either the square grid $\mathbb{Z}^2$ or an infinite binary tr...

If an infinite graph is $C$-roughly transitive for some finite $C$, must it be roughly isometric to a homogeneous metric space? Equivalently, does the...

For every Cayley graph $G$, does there exist $r=r(G)$ such that $G$ covers every graph whose radius-$r$ balls are all isomorphic to the radius-$r$ bal...

Fix a rooted radius-$r$ ball $B(o,r)$ that occurs as every radius-$r$ ball of some finite graph. What is the minimum diameter of a finite graph all of...

Let $G$ be an $n$-vertex graph whose rooted balls of size $k$ are all isomorphic. If $k>n/2$, or if $k$ is within a fixed constant of $\operatorname{d...

If an infinite vertex-transitive graph is $f(r)$-sofic for some function $f$, must it be $cr$-sofic for a constant $c$? More uniformly, for fixed degr...

Prove that no infinite connected Ramanujan graph is Liouville; equivalently, every such graph admits a nonconstant bounded harmonic function....

Prove that every bounded-degree graph roughly isometric to a nonamenable Cayley graph is non-Liouville....

Suppose $\mathbb{Z}$ acts on a graph $G$ by isometries, the quotient $H=G/\mathbb{Z}$ is Liouville, and simple random walk on $G$ visits every transla...

Let $(G_k)$ be a locally convergent sequence of bounded-degree graphs, each having separation profile of order at most the square root of the subgraph...

Prove that there is a universal constant $C$ such that every finite connected vertex-transitive graph $G$ of degree $d$ satisfies $$R_{\mathrm{eff}}(u...

Does there exist a family $(G_n)$ of finite $d$-regular graphs with $|G_n|\to\infty$ and a constant $h>0$ such that every induced metric ball in every...