
TL;DR
This paper investigates the distribution of prime differences, establishing new bounds on intervals containing numbers that are differences of two primes, and discusses conditional results based on unproven hypotheses about prime distribution.
Contribution
It introduces a new bound showing that intervals of the form [X, X+(logX)^C] always contain a difference of two primes for large X, and proves a weaker result for smaller intervals.
Findings
Intervals of the form [X, X+(logX)^C] contain differences of two primes for large X.
For any small c>0, [X, X+X^c] contains a difference of two primes when X is large.
Conditional results depend on hypotheses about primes' distribution level.
Abstract
In the present work we investigate the largest possible gaps between consecutive numbers which can be written as the difference of two primes. The best known upper bounds are the same as those concerning the largest possible difference of Goldbach numbers (that is, numbers which can be written as the sum of two primes). Thus, we know that any interval of the form [X, X+X^c] contains numbers which are the difference (or sum, respectively) of two primes, where c=21/800. It is announced in our work that there is a constant C such that for sufficiently large X all intervals of the form [X, X+(logX)^C] contain an even integer which can be written as the difference of two primes. The work contains, as an illustration of the method, the proof of the weaker result that given an arbitrarily small c>0, the interval [X, X+X^c] contains the difference of two primes if X is large enough. Some…
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Taxonomy
TopicsAnalytic Number Theory Research · Advanced Mathematical Identities · Benford’s Law and Fraud Detection
