  There exists a Hamiltonian formulation of the factorisation problem which
also needs the definition of a factorisation ensemble (a set to which
factorable numbers, $N'=x'y'$, having the same trivial factorisation
algorithmic complexity, belong). For the primes therein, a function $E$, that
may take only discrete values, should be the analogous of the energy from a
confined system of charges in a magnetic trap. This is the quantum factoring
simulator hypothesis connecting quantum mechanics with number theory. In this
work, we report numerical evidence of the existence of this kind of discrete
spectrum from the statistical analysis of the values of $E$ in a sample of
random OpenSSL n-bits moduli (which may be taken as a part of the factorisation
ensemble). Here, we show that the unfolded distance probability of these $E$'s
fits to a {\it Gaussian Unitary Ensemble}, consistently as required, if they
actually correspond to the quantum energy levels spacing of a magnetically
confined system that exhibits chaos. The confirmation of these predictions
bears out the quantum simulator hypothesis and, thereby, it points to the
existence of a liaison between quantum mechanics and number theory. Shor's
polynomial time complexity of the quantum factorisation problem, from pure
quantum simulation primitives, was obtained.
