ABSTRACT
The problem of modeling the distribution of prime numbers is a popular subject of investigation for number theorists. When looking at individual numbers, the primes seem to be randomly distributed, but the ‘‘global’’ distribution of primes follows well-defined laws [1, 2]. Let:
d = Pi+1 − Pi, i ∈ N (21.1) be the distance between consecutive prime numbers. An algorithm based on the sliding window technique was applied in order to find out n-tuples of neighbor distances within some range of prime numbers, i.e.:
{d1, . . . , dk , . . . , dn} ⊂ max J , k , n ∈ N dk = Pk+1 − Pk
(21.2)
where J is a counting function for each n-tuple, this function rends only n-tuples having maximal count. Obviously, shorter n-tuples produce larger collection of n-tuples that can be found. In particular, since 6 is the distance which can be found within primes of order up to 1035 [3], then the couple {6, 6} will be the most common couple of neighbor distances for that range, i.e.:
max{Pi+1 − Pi, Pi+2 − Pi+1} ⇔ (Pi+1 − Pi = 6 ∧ Pi+2 − Pi+1 = 6) (21.3) A record can be kept of the indices for the primes involved in the generation of each couple, and
it could be conjectured that there might be some regularity or some sort of pattern in the distribution of those indices. In particular let us choose the indices i, i+2 so that:
j ︷ ︸︸ ︷ {i, i + 2},
j+1 ︷ ︸︸ ︷ {i + k , i + k + 2} k ∈ N (21.4)
Each k is unknown a priori, since the exact position of each couple is not known, unless of course the distribution of the prime numbers is known with full precision [2]. However one could try to find out something about that set through the distances between some of the elements of each couple, so for example let:
md = (i + kj+1) − (i + kj+2) md ∈ Md (21.5) where md stands as shorthand for ‘meta-distances’, implying distances embedded in the set of distances between prime numbers, Md is the set of all the meta-distances within a given range of prime numbers, and the way they are obtained is the difference between the first element of indices for a given set, minus the second element of the previous set. Unfortunately the algorithms so far applied to the set Md have rendered inconclusive results about a possible pattern [4]. However, it
for integration, as are Monte Carlo methods [4]. Also the generation of a probability density function (pdf) which could describe the local distribution of a given subset of Md was researched [4]. For the generation of the pdf, a collection of bins was created, each one ranging from some lower bound to an upper bound:
bl = n∑
i=1 [mdk , mdr], r > k b1 < · · · < bl < · · · < bL (21.6)
Then a set of parameters A, B ∈ R, can be chosen through a non-linear fit, using the LevenbergMarquardt algorithm, to render:
pdf = f (bl) = A exp(−Bx2) (21.7)
Here, how the parameters A, B change for every subset of Md, each subset coming from 106 prime numbers has been researched. Due to hardware limitations, this was done only for the first 43 of such subsets. It is also conjectured that Md are a low-discrepancy sequence, thus making Md useful for integration. Later it will be shown that Md perform nearly as well for Monte Carlo simulations, thus partially agreeing with [5], in that low-discrepancy sequences are good for multivariate integration, since they occupy the hyperspace uniformly, but not that useful for Monte Carlo simulations, such as calculations in thermodynamics.
