Sequences from number theory for physics,signal processing,and art

In this article I discuss numerical sequences of potential interest to physicists, engineers, and artists. Foremost among these are the maximum-length or “Galois” sequences, based on the prime number 2 and larger primes, which have found wide use in physics, including acoustics, and engineering:

• —the measurement of impulse responses in concert halls, radar echoes from planets (to check the General Theory of Relativity), and travel times in the deep-ocean sound channel (to monitor water temperature and global warming)
• —the spatial diffusion of sound waves, coherent (laser) light, and electromagnetic waves
• —algebraic error-correcting codes (Simplex and Hamming codes)
• —minimizing peak-factors for radar and sonar signals, synthetic speech, and computer music
• —the formation of X-ray images with 2D masks (in X-ray astronomy).

Other sequences include

• —quadratic-residue sequences for the construction of wide-band diffusing reflection-phase-gratings in one and two dimensions
• —the Morse-Thue sequence, the Fibonacci and “rabbit” sequences and their musical potential; and
• —certain self-similar sequences from number theory that engender attractive visual patterns, rhythms, and melodies.