What are the transitive permutation groups of prime (or prime power) degree?
Are there infinitely many simple groups of order a product of six primes?
Are there infinitely many counterexamples to a theorem of Cauchy on permutation groups?

Solving these and various problems in other areas of mathematics, such as the twin primes conjecture, depends on certain conjectures in Number Theory regarding prime values of polynomials, namely Schinzel's Hypothesis and its quantified form, the Bateman-Horn Conjecture. Proving these is a very difficult open problem, but joint work with Alexander Zvonkin (LaBRI, Bordeaux) gives strong computational evidence that in the above contexts (and many others) they are true.

(No background in Number Theory is required for this talk, beyond knowing the definition of a prime number, and a few examples. Similarly, only elementary Group Theory is required.)