Computational Model of Prime Numbers by the Modified Chi-square Function

Abstract

An innovative approach that treats prime numbers as raw experimental data and as elements of larger and larger finite sequences {Pm}≡{P(mp)} is shown in the present report. The modified chi-square function Xk2(A,mp/xo) with its three parameters A, k and xo=xo(k) is the best-fit function of the finite sequences {ρm}≡{lgPm/lgmp} from the analytical viewpoint thus showing that the property of scale invariance does not hold for the finite sequences of this prime variable and so for primes themselves. In addition an injective map can be set between these {ρm} sequences and the {mα} progressions with domain N and co-domain R+ being α∈(–1,0)⊂R– through the parameter k=2+2α of their common fit function Xk2(A,mp/xo). All that leads to induction algorithms and to relationships of the kind Pm≈P(mp), though within the precisions of the calculations and holding locally.