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The set of primes where a hypergeomeric series with rational parameters is$p$-adically bounded is known by [10] to have a Dirichlet density. We establisha formula for this Dirichlet density and conjecture that it is rare for thedensity to be large. We prove this conjecture for hypergeometric series whoseparameters have denominators equal to a prime of the form $p = 2q^r + 1$, where$q$ is an odd prime, by establishing an upper bound on the density of boundedprimes in this case. The general case remains open. This paper is the output ofan undergraduate research course taught by the first listed author in thewinter semester of 2018.

Densities of bounded primes for hypergeometric series with rational parameters