prime numbers

[Rothiemurchus]

Are prime numbers more than a curiosity? I know they can be useful in encrypting data but do they have a more fundamental role in the physical world? For example,a prime number won't split into two equal integer numbers.
Integers occur in quantum mechanics - our most fundamental description of particles.Do primes occur because particles don't like to split into equal integer fragments?

[matt grime]

Distribution of primes related to energy levels of some dynamical systems. See Random matrix theory.

Primes intricately related to possible group structures - elementary particles behave as groups.