Prey population cycles are stable in an evolutionary model if and only if their periods are prime

We present an evolutionary model that marks an encounter of two seemingly unrelated disciplines: population dynamics and number theory. Assuming mutations and selection of predators and prey, we show that prey cycles with non-prime lengths are unstable, while cycles with prime lenghts are stable. Allowing arbitrarily long cycles, this model is a number-theoretical tool for the calculation of large prime numbers. An extension of this purely temporal process to an evolutionary game on a spatial array leads to homogeneity, or to travelling or spiral waves having a predominance of prime prey cycle lengths. These results may be related to the appearance of cicadas (genus Magicicadae) every 13 or 17 years.