Some experiments in number theory

Following footsteps of Gauss, Euler, Riemann, Hurwitz, Smith, Hardy, Littlewood, Hedlund, Khinchin and Chebyshev, we visit some topics in elementary number theory. For matrices defined by Gaussian primes we observe a circular spectral law for the eigenvalues. We experiment then with various Goldbach conjectures for Gaussian primes, Eisenstein primes, Hurwitz primes or Octavian primes. These conjectures relate with Landau or Bunyakovsky or Andrica type conjectures for rational primes. The Landau problem asking whether infinitely many predecessors of primes are square is also related to a determinant problem for the prime matrices under consideration. Some of these matrices are adjacency matrices of bipartite graphs. Their Euler characteristics in turn is related to the prime counting function. When doing statistics of Gaussian primes on rows, we detect a sign of correlations: rows of even distance for example look asymptotically correlated. The expectation values of prime densities were conjectured to converge by Hardy-Littlewood almost 100 years ago. We probe the convergence to these constants, following early experimenters. After factoring out the dihedral symmetry of Gaussian primes, they are bijectively related to the standard primes but the sequence of angles appears random. A similar story happens for Eisenstein primes. Gaussian or Eisenstein primes have now a unique angle attached to them. We also look at the eigenvalue distribution of greatest common divisor matrices whose explicitly known determinants are given number theoretically by Jacobi totient functions and where unexplained spiral patterns can appear in the spectrum. Related are a class of graphs for which the vertex degree density is related to the Euler summatory totient function. We then apply cellular automata maps on prime configurations. Examples are Conway’s life and moat-detecting cellular automata which we ran on Gaussian primes. Related to prime twin conjectures and more general pattern conjectures for Gaussian primes is the question whether ”life” exists arbitrary far away from the origin, even if is primitive life in form of a blinker obtained from a prime twin. Most questions about Gaussian primes can be asked for Hurwitz primes inside the quaternions, for which the zeta function is just shifted. There is a Goldbach statement for quaternions: we see experimentally that every Lipschitz integer with entries larger than 1 is a sum of two Hurwitz primes with positive entries and every Hurwitz prime with entries larger than 3 is a sum of a Hurwitz and Lipschitz prime. For Eisenstein primes, we see that all but finitely many Eisenstein integers with coordinates larger than 2 can be written as a sum of two Eisenstein primes with positive coordinates. We also predict that every Eisenstein integer is the sum of two Eisenstein primes without any further assumption. For coordinates larger than 1, there are two curious ghost examples. For Octonions, we see that there are arbitrary large Gravesian integer with entries larger than 1 which are not the sum of two Kleinian primes with positive coordinates but we ask whether every Octavian integers larger than some constant K is a sum of two Octavian primes with positive coordinates. Finally we look at some spectra of almost periodic pseudo random matrices defined by Diophantine irrational rotations, where fractal spectral phenomena occur. The matrix is the real part of a van der Monde matrix whose determinant has relations with the curlicue problem in complex analysis or the theory of partitions of integers. Diophantine properties allow to estimate the growth rate of the determinants of these complex matrices if the rotation number is the golden mean.