# Ulam, Stanislaw M.

## Are there an infinite number of lucky primes? ★

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

**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.

## Dense rational distance sets in the plane ★★★

Author(s): Ulam

**Problem**Does there exist a dense set so that all pairwise distances between points in are rational?

Keywords: integral distance; rational distance

## Reconstruction conjecture ★★★★

The *deck* of a graph is the multiset consisting of all unlabelled subgraphs obtained from by deleting a vertex in all possible ways (counted according to multiplicity).

**Conjecture**If two graphs on vertices have the same deck, then they are isomorphic.

Keywords: reconstruction