Ulam, Stanislaw M.

Are there an infinite number of lucky primes?

Author(s): Lazarus: Gardiner: Metropolis; Ulam

\begin{conjecture} If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime. \end{conjecture}

Keywords: lucky; prime; seive

Dense rational distance sets in the plane ★★★

Author(s): Ulam

\begin{problem} Does there exist a dense set $S \subseteq {\mathbb R}^2$ so that all pairwise distances between points in $S$ are rational? \end{problem}

Keywords: integral distance; rational distance

Reconstruction conjecture ★★★★

Author(s): Kelly; Ulam

The \emph{deck} of a graph $G$ is the multiset consisting of all unlabelled subgraphs obtained from $G$ by deleting a vertex in all possible ways (counted according to multiplicity).

\begin{conjecture} If two graphs on $\ge 3$ vertices have the same deck, then they are isomorphic. \end{conjecture}

Keywords: reconstruction

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