The Distribution of Primes: Patterns in Chaos?

Know­ing there are in­fi­nite­ly many primes, math­e­mati­cians nat­u­ral­ly asked: how are they dis­trib­uted among the in­te­gers? Are they common or rare? Do they follow any pat­tern?

Gaps Be­tween Primes

Primes can be sur­pris­ing­ly close to­geth­er (like 3, 5, 7 or 11, 13). How­ev­er, one can also find ar­bi­trar­i­ly large gaps be­tween con­sec­u­tive primes. For any pos­i­tive in­te­ger $k$, we can find a se­quence of $k$ con­sec­u­tive com­pos­ite in­te­gers. Con­sid­er the se­quence:

Let's an­a­lyze these terms:

• $(k+1)! + 2$ is di­vis­i­ble by 2 (since $(k+1)!$ con­tains 2 as a factor).

• $(k+1)! + 3$ is di­vis­i­ble by 3 (since $(k+1)!$ con­tains 3 as a factor, as­sum­ing $k \ge 2$).

• ...

• $(k+1)! + (k+1)$ is di­vis­i­ble by $(k+1)$.

Each number in this se­quence of $k$ in­te­gers is com­pos­ite. This means that while primes are in­fi­nite, they can become very sparse.

Twin Primes

Twin primes are pairs of prime num­bers that differ by 2.

Ex­am­ples in­clude:

• (3, 5)

• (5, 7)

• (11, 13)

• (17, 19)

• (29, 31)

A major un­solved ques­tion in number theory is the Twin Prime Con­jec­ture, which states that there are in­fi­nite­ly many twin prime pairs. While not yet proven, sig­nif­i­cant progress has been made in recent years, sug­gest­ing it is likely true.

The Prime Number The­o­rem (PNT)

While the local dis­tri­b­u­tion of primes can seem chaot­ic, their global dis­tri­b­u­tion ex­hibits a re­mark­able reg­u­lar­i­ty. This is de­scribed by the Prime Number The­o­rem (PNT).

Let $\pi(x)$ (pro­nounced "pi of x") be the prime-count­ing func­tion, which de­notes the number of prime num­bers less than or equal to $x$.

For ex­am­ple:

• $\pi(10) = 4$ (primes are 2, 3, 5, 7)

• $\pi(30) = 10$ (primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29)

The Prime Number The­o­rem states that $\pi(x)$ is as­ymp­tot­i­cal­ly equiv­a­lent to $\frac{x}{\ln x}$:

Here, $\ln x$ is the nat­ur­al log­a­rithm of $x$, and the symbol $\sim$ means that the ratio of the two func­tions ap­proach­es 1 as $x$ ap­proach­es in­fin­i­ty: $\lim_{x \to \infty} \frac{\pi(x)}{x / \ln x} = 1$.

This the­o­rem tells us that, on av­er­age, the den­si­ty of primes around a number $x$ is about $1/\ln x$. So, primes become less common as num­bers get larger, but they don't thin out too quick­ly.

A more ac­cu­rate ap­prox­i­ma­tion for $\pi(x)$ is given by the log­a­rith­mic in­te­gral func­tion, de­not­ed $\text{Li}(x)$ (pro­nounced "ell-eye of x" or "log­a­rith­mic in­te­gral of x"), de­fined as:

The PNT can also be stated as $\pi(x) \sim \text{Li}(x)$. This ap­prox­i­ma­tion is gen­er­al­ly much better than $x/\ln x$.

The PNT was con­jec­tured by Le­gendre and Gauss in the late 18th/early 19th cen­tu­ry based on nu­mer­i­cal ev­i­dence. It was in­de­pen­dent­ly proven by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896, using com­plex analy­sis, par­tic­u­lar­ly prop­er­ties of the Rie­mann zeta func­tion.