Special Types of Primes and Famous Conjectures

Beyond the gen­er­al study of primes, cer­tain spe­cif­ic forms of primes and re­lat­ed con­jec­tures have at­tract­ed much at­ten­tion.

Mersenne Primes

Mersenne primes are primes of the form $M_p = 2^p - 1$, where $p$ itself must be a prime number. (If $p$ were com­pos­ite, say $p=ab$, then $2^{ab}-1 = (2^a-1)(1+2^a+\ldots+2^{a(b-1)})$, so $2^p-1$ would be com­pos­ite.) Ex­am­ples:

• $M_2 = 2^2 - 1 = 3$ (prime)

• $M_3 = 2^3 - 1 = 7$ (prime)

• $M_5 = 2^5 - 1 = 31$ (prime)

• $M_7 = 2^7 - 1 = 127$ (prime)

• $M_{11} = 2^{11} - 1 = 2047 = 23 \cdot 89$ (not prime, so $p$ being prime is nec­es­sary but not suf­fi­cient)

Mersenne primes are linked to per­fect num­bers (num­bers equal to the sum of their proper di­vi­sors, e.g., $6 = 1+2+3$). If $M_p$ is a Mersenne prime, then $2^{p-1}M_p$ is an even per­fect number (Euclid-Euler the­o­rem). The search for new Mersenne primes is an active area, often pur­sued by dis­trib­uted com­put­ing projects like GIMPS (Great In­ter­net Mersenne Prime Search). As of late 2023, 51 Mersenne primes are known, the largest being $2^{82,589,933} - 1$.

Fermat Primes

Fermat primes are primes of the form $F_n = 2^{(2^n)} + 1$. Pierre de Fermat con­jec­tured that all num­bers of this form were prime.

• $F_0 = 2^{(2^0)} + 1 = 2^1 + 1 = 3$ (prime)

• $F_1 = 2^{(2^1)} + 1 = 2^2 + 1 = 5$ (prime)

• $F_2 = 2^{(2^2)} + 1 = 2^4 + 1 = 17$ (prime)

• $F_3 = 2^{(2^3)} + 1 = 2^8 + 1 = 257$ (prime)

• $F_4 = 2^{(2^4)} + 1 = 2^{16} + 1 = 65,537$ (prime)

How­ev­er, Euler showed in 1732 that $F_5 = 2^{32} + 1 = 4,294,967,297 = 641 \cdot 6,700,417$, so $F_5$ is com­pos­ite. In fact, no Fermat num­bers $F_n$ for $n > 4$ are known to be prime. Fermat primes are con­nect­ed to the con­structibil­i­ty of reg­u­lar poly­gons with a com­pass and straight­edge. Gauss proved that a reg­u­lar $N$-gon can be con­struct­ed if and only if the odd prime fac­tors of $N$ are dis­tinct Fermat primes.

Sophie Ger­main Primes

A prime number $p$ is a Sophie Ger­main prime (named after French math­e­mati­cian Sophie Ger­main) if $2p+1$ is also prime. The number $2p+1$ as­so­ci­at­ed with a Sophie Ger­main prime is called a safe prime.

Ex­am­ples:

• If $p=2$, $2p+1 = 5$ (prime). So 2 is a Sophie Ger­main prime.

• If $p=3$, $2p+1 = 7$ (prime). So 3 is a Sophie Ger­main prime.

• If $p=5$, $2p+1 = 11$ (prime). So 5 is a Sophie Ger­main prime.

• If $p=11$, $2p+1 = 23$ (prime). So 11 is a Sophie Ger­main prime.

It is con­jec­tured that there are in­fi­nite­ly many Sophie Ger­main primes, but this is un­proven.

The Gold­bach Con­jec­ture

One of the oldest and best-known un­solved prob­lems in number theory. Pro­posed by Chris­t­ian Gold­bach in a 1742 letter to Euler, it has two common forms:

• Strong Gold­bach Con­jec­ture: Every even in­te­ger greater than 2 is the sum of two primes. (e.g., $4=2+2$, $6=3+3$, $8=3+5$, $20=3+17=7+13$)

• Weak Gold­bach Con­jec­ture: Every odd in­te­ger greater than 5 is the sum of three primes. (The weak con­jec­ture was proven by Harald Helf­gott in 2013. The strong con­jec­ture im­plies the weak one.) The strong Gold­bach Con­jec­ture has been ver­i­fied for num­bers up to $4 \times 10^{18}$ but re­mains un­proven.

Le­gendre's Con­jec­ture

Pro­posed by Adrien-Marie Le­gendre, this con­jec­ture states that there is always a prime number be­tween $n^2$ and $(n+1)^2$ for every pos­i­tive in­te­ger $n$. (e.g., for $n=1$: be­tween $1^2=1$ and $2^2=4$, primes are 2, 3. For $n=2$: be­tween $2^2=4$ and $3^2=9$, primes are 5, 7.) This is also un­proven. It is one of Landau's prob­lems.

Landau's Prob­lems

At the 1912 In­ter­na­tion­al Con­gress of Math­e­mati­cians, Edmund Landau listed four basic prob­lems about prime num­bers that were "un­at­tack­able at the present state of sci­ence." They are:

1. Gold­bach's Con­jec­ture

2. Twin Prime Con­jec­ture

3. Le­gendre's Con­jec­ture

4. Are there in­fi­nite­ly many primes of the form $n^2+1$? (e.g., $2^2+1=5$, $4^2+1=17$, $6^2+1=37$) All four prob­lems remain un­solved over a cen­tu­ry later.