Prime suspects and lucky numbers

Prof. Giuseppe Mussardo (SISSA)

Abstract

Prime numbers play a crucial role in mathematics being the key elements for the factorization of the integers. The idea to use them for designing a quantum abacus has recently received a new support from the experimental realization of a single-particle quantum Schrodinger Hamiltonian whose eigenvalues are given by the first N prime numbers. Such an experimental set-up consists of light intensities profiles, tuned by a computer-generated holography, able to create an optical trap for ultracold atoms. The statistical properties of the primes, such as their asymptotic scaling law for the n-th prime p! ≃ n log n, besides being the key to implement such a quantum potential, are also shared by other sequences of integers obtained by means of sieves. This is the case of the so-called "lucky numbers”, originally studied by Stan Ulam, for which there exists indeed an associated quantum Hamiltonian. Integrated with other considerations, these two examples pave the way toward the possibility to set up quantum systems able of performing arithmetic manipulations, including the factorization of integers.