Subtleties in the Distribution of Primes
There are only very few proved results concerning the distribution of primes in short intervals. The prime number theorem tells us that the average density of primes around x is approximately 1/In x. This means that if we consider an interval of length Δx about x and choose any integer t in this interval, then the probability of t being a prime will approach 1/ In x as x → ∞, if Δx is small compared to x. This implies that the primes tend to thin out as x grows larger; an implication that becomes obvious when considering that the condition for a randomly picked integer x to be composite is that it has some prime factor \( \leqslant \sqrt x\) and that there are more prime factors \( \leqslant \sqrt x\) to choose from when x is larger.
KeywordsNumber Series Residue Class White Ball Infinite Product Consecutive Integer
- 3.F. J. Gruenberger and G. Armerding, Statistics on the First Six Million Prime Numbers, Reviewed in Math. Comp. 19 (1965) pp. 503–505.Google Scholar
- 8.Thomas Vehka, “Explicit Construction of an Admissible Set for the Conjecture that Sometimes π(x + y) > π(y) + π(x)” Notices Am. Math. Soc. 26 (1979) p. A-453.Google Scholar
- 19.Harald Cramér, “On the Order of Magnitude of the Difference Between Consecutive Prime Numbers,” Acta Arith. 2 (1936) pp. 23–46.Google Scholar