Abstract
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/ln 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/ln 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.
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Riesel, H. (1994). Subtleties in the Distribution of Primes. In: Prime Numbers and Computer Methods for Factorization. Progress in Mathematics, vol 126. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0251-6_3
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DOI: https://doi.org/10.1007/978-1-4612-0251-6_3
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