Samuel Gross (Rocky Mountain College)


Let $f(x)$ be a polynomial with non-negative integer coefficients for which $f(10)$ is a prime. A result of A. Cohn implies that if the coefficients of $f(x)$ are $\le 9$, then $f(x)$ is irreducible. In 1988, M. Filaseta showed that the bound $9$ could be replaced by $10^{30}$. Can we do better? We will answer this question, discuss other numbers similar to the one in the title of this talk, and explore some open problems related to our recent work.