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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.

The goal of discrete tomography is to reconstruct an unknown function f via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of switching functions, or ghosts, which allow many solutions to exist in general. Previous work has shown that it is possible to determine all solutions in linear time (with respect to the number of directions and grid size) regardless of whether the solution is unique. In this work, we show that a similar linear algorithm exists in three dimensions. This is achieved by viewing the three-dimensional grid along each 2D coordinate plane, effectively solving the problem with a series of 2D linear algorithms. By that, it is possible to solve the problem of 3D discrete tomography in linear time.

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