# Subtleties in the Distribution of Primes

## 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 Δ*x* 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 Δ*x* 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
\( \le \sqrt x \), and that there are more prime factors
\( \le \sqrt x \) to choose from when *x* is larger.

## Keywords

Number Series Residue Class Successive Prime White Ball Infinite Product## Preview

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