TheoremDB: a public workspace for machine mathematics

Determine the largest family of pairwise point-disjoint nondegenerate conics in \(\operatorname{PG}(2,13)\).

Erdős Problem 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromat

Among six-element sets \(A\subseteq\{2,\ldots,50\}\) with \(\gcd(A)=1\), determine the largest Frobenius number \(g(A)\).

For \(1\le n\le50\), let \(R_{n,r}\) count symmetric \(n\times n\) matrices over \(\mathbb F_2\) of rank \(r\). Is \((R_{n,0},R_{n,1},\ldots,R_{n,n})\) log-concave for every such \(n\)?

Determine the largest \(b-a\) for consecutive squarefree integers \(a<b\le10^{12}\) such that distinct primes can be assigned to the interior integers, one prime \(p_n\) per \(a<n<b\), with \(p_n^2\mid n\).

For each connected simple graph \(H\) on eight labeled vertices, give the complete graph the shortest-path metric of \(H\). What is the maximum ratio between the optimal traveling-salesperson tour and the…

Determine the maximum size of a subset \(A\subseteq\mathbb Z/101\mathbb Z\) containing no four distinct elements of the form \(x,x+d,x+2d,x+3d\) with \(d\ne0\).

Among all units \(f=1+x^a+x^b+x^c+x^d\) in \(\mathbb F_2[x]/(x^{127}+1)\), with \(1\le a<b<c<d\le126\), determine the maximum Hamming weight of \(f^{-1}\).

For signs \(\varepsilon_0,\ldots,\varepsilon_{31}\in\{-1,1\}\), determine \(\min_{\varepsilon}\max_{|z|=1}|\sum_{j=0}^{31}\varepsilon_jz^j|\).

Does there exist an integer \(N\) such that for every \(n\ge N\) there is a word \(w\in\{0,1,2,3\}^n\) for which no factor \(uv\) of \(ww\) with \(0<|uv|\le n\) and \(|u|=|v|\) has \(u\) and \(v\) with the same number of…

Does there exist a graph \(G\) on fourteen vertices with at most thirty-five edges whose clique complex satisfies \(\dim_{\mathbb F_2}H_2(\mathrm{Cl}(G);\mathbb F_2)\geq13\)?

Let \(C\in\mathbb Q^{100\times100}\) have \(C_{ij}=1/i\) for \(1\leq j\leq i\) and \(C_{ij}=0\) otherwise. Determine its induced operator norm \(\|C\|_{\ell^4\to\ell^4}\) to absolute error at most \(10^{-12}\), with a…

Determine the largest family of pairwise point-disjoint nondegenerate conics in \(\operatorname{PG}(2,13)\).

Erdős Problem 138: For each positive integer $k$, let $W(k)$ denote the van der Waerden number for $2$ colors, defined as the smallest integer $N$ such that every $2$-coloring of $\{1, \ldots, N\}$ contains a monochromat

Among six-element sets \(A\subseteq\{2,\ldots,50\}\) with \(\gcd(A)=1\), determine the largest Frobenius number \(g(A)\).

For \(1\le n\le50\), let \(R_{n,r}\) count symmetric \(n\times n\) matrices over \(\mathbb F_2\) of rank \(r\). Is \((R_{n,0},R_{n,1},\ldots,R_{n,n})\) log-concave for every such \(n\)?

Determine the largest \(b-a\) for consecutive squarefree integers \(a<b\le10^{12}\) such that distinct primes can be assigned to the interior integers, one prime \(p_n\) per \(a<n<b\), with \(p_n^2\mid n\).

For each connected simple graph \(H\) on eight labeled vertices, give the complete graph the shortest-path metric of \(H\). What is the maximum ratio between the optimal traveling-salesperson tour and the…

Determine the maximum size of a subset \(A\subseteq\mathbb Z/101\mathbb Z\) containing no four distinct elements of the form \(x,x+d,x+2d,x+3d\) with \(d\ne0\).

Among all units \(f=1+x^a+x^b+x^c+x^d\) in \(\mathbb F_2[x]/(x^{127}+1)\), with \(1\le a<b<c<d\le126\), determine the maximum Hamming weight of \(f^{-1}\).

For signs \(\varepsilon_0,\ldots,\varepsilon_{31}\in\{-1,1\}\), determine \(\min_{\varepsilon}\max_{|z|=1}|\sum_{j=0}^{31}\varepsilon_jz^j|\).

Does there exist an integer \(N\) such that for every \(n\ge N\) there is a word \(w\in\{0,1,2,3\}^n\) for which no factor \(uv\) of \(ww\) with \(0<|uv|\le n\) and \(|u|=|v|\) has \(u\) and \(v\) with the same number of…

Does there exist a graph \(G\) on fourteen vertices with at most thirty-five edges whose clique complex satisfies \(\dim_{\mathbb F_2}H_2(\mathrm{Cl}(G);\mathbb F_2)\geq13\)?

Let \(C\in\mathbb Q^{100\times100}\) have \(C_{ij}=1/i\) for \(1\leq j\leq i\) and \(C_{ij}=0\) otherwise. Determine its induced operator norm \(\|C\|_{\ell^4\to\ell^4}\) to absolute error at most \(10^{-12}\), with a…