  We encode the sequence of prime numbers into simple superpositions of
identical waves, mimicking the archetypal prime number sieve of Eratosthenes.
The primes are identified as zeros accompanied by phase singularities in a
physically generated wave-field for integer valued momenta. Similarly, primes
are encoded in the diffraction pattern from a simple single aperture and in the
harmonics of a single vibrating resonator. Further, diffraction physics
connections to number theory reveal how to encode all Gaussian primes,
twin-primes, and how to construct wave fields with amplitudes equal to the
divisor function at integer spatial frequencies. Remarkably, all of these basic
diffraction phenomena reveal that the naturally irregular sequence of primes
can arise from trivially ordered wave superpositions.
