TheoremDB

Every smooth closed four-manifold \(M\) that is homotopy equivalent to \(S^4\) is diffeomorphic to the standard smooth four-sphere \(S^4\).

Does there exist an infinite group \(G\) of exponent \(5\) such that every nontrivial proper subgroup of \(G\) is cyclic of order \(5\)?

For the ferromagnetic Ising model on \(C_6\mathbin{\square}C_6\) at inverse temperature \(\beta=(\log2)/2\) and zero field, choose one vertex uniformly and resample its spin from the conditional Gibbs law. Determine the…

For \(d\ge 2\), \(1<p<\infty\), and \(\delta>\max\{d|1/p-1/2|-1/2,0\}\), are the Euclidean Bochner-Riesz multipliers \(S_R^\delta\), defined by the Fourier multiplier \((1-|\xi|^2/R^2)_+^\delta\), bounded on…

Erdős Problem 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here,

For every \(\varepsilon,\delta>0\), there exists an alphabet size \(q\) such that it is NP-hard to distinguish unique games with optimum at least \(1-\varepsilon\) from those with optimum at most \(\delta\).

If a strictly convex \(C^\infty\) planar billiard table has an invariant essential caustic for every rotation number \(\rho\in(0,\tfrac12)\), must its boundary be an ellipse?

If a nontrivial knot \(K\subset S^3\) has a Dehn surgery producing a reducible three-manifold, must \(K\) be a cable knot and must the surgery slope be its cabling slope?

Define a sequence \((a_m)_{m\ge 1}\) of positive integers recursively by \(a_1=1\), and, for each \(m\ge 2\), let \(a_m\) be the least positive integer such that \(a_{m-2k}+a_m\ne 2a_{m-k}\) for every integer \(k\) with…

Process the vertices of the \(8\times8\) grid graph in a uniformly random order, accepting a vertex exactly when none of its previously accepted neighbors is present. What is the exact expected final number \(\mathbb…

For every \(\varepsilon>0\), does there exist \(k\ge3\) such that \(k\)-SAT on \(n\) variables cannot be decided in time \(O((2-\varepsilon)^n)\) by a deterministic algorithm?

There are infinitely many primes \(p\) for which \(p+2\) is also prime.

Every smooth closed four-manifold \(M\) that is homotopy equivalent to \(S^4\) is diffeomorphic to the standard smooth four-sphere \(S^4\).

Does there exist an infinite group \(G\) of exponent \(5\) such that every nontrivial proper subgroup of \(G\) is cyclic of order \(5\)?

For the ferromagnetic Ising model on \(C_6\mathbin{\square}C_6\) at inverse temperature \(\beta=(\log2)/2\) and zero field, choose one vertex uniformly and resample its spin from the conditional Gibbs law. Determine the…

For \(d\ge 2\), \(1<p<\infty\), and \(\delta>\max\{d|1/p-1/2|-1/2,0\}\), are the Euclidean Bochner-Riesz multipliers \(S_R^\delta\), defined by the Fourier multiplier \((1-|\xi|^2/R^2)_+^\delta\), bounded on…

Erdős Problem 82: For each positive integer \(n\), let \(F(n)\) denote the largest integer such that every finite simple graph on \(n\) vertices contains a regular induced subgraph with at least \(F(n)\) vertices. Here,

For every \(\varepsilon,\delta>0\), there exists an alphabet size \(q\) such that it is NP-hard to distinguish unique games with optimum at least \(1-\varepsilon\) from those with optimum at most \(\delta\).

If a strictly convex \(C^\infty\) planar billiard table has an invariant essential caustic for every rotation number \(\rho\in(0,\tfrac12)\), must its boundary be an ellipse?

If a nontrivial knot \(K\subset S^3\) has a Dehn surgery producing a reducible three-manifold, must \(K\) be a cable knot and must the surgery slope be its cabling slope?

Define a sequence \((a_m)_{m\ge 1}\) of positive integers recursively by \(a_1=1\), and, for each \(m\ge 2\), let \(a_m\) be the least positive integer such that \(a_{m-2k}+a_m\ne 2a_{m-k}\) for every integer \(k\) with…

Process the vertices of the \(8\times8\) grid graph in a uniformly random order, accepting a vertex exactly when none of its previously accepted neighbors is present. What is the exact expected final number \(\mathbb…

For every \(\varepsilon>0\), does there exist \(k\ge3\) such that \(k\)-SAT on \(n\) variables cannot be decided in time \(O((2-\varepsilon)^n)\) by a deterministic algorithm?

There are infinitely many primes \(p\) for which \(p+2\) is also prime.