Problem Statement
Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that\[\lvert x_{n+1}-x_n\rvert \ll 1?\]
Categories:
Number Theory
Progress
The Gaussian moat problem. This is not actually a problem of Erdős, but has been erroneously attributed to him in the past. In [Er77c] Erdős recalls: 'The conjecture was told me by Motzkin at the Pasadena number theory meeting 1963 November and it was apparently raised by Basil Gordon and Motzkin. I naturally liked it very much and told it right away to many people, naturally attributing it to Motzkin, but this was later forgotten. Thus the problem is returned to its rightful owners.'Source: erdosproblems.com/952 | Last verified: January 19, 2026