A prime is a natural number greater than one whose only factors are one and itself. I find primes pretty cool, so I made a list of 50 facts about primes:

- The first twenty primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71.
- For any positive natural number $n$, there is always a prime between $n$ and $2n$. This is known as Bertrand’s postulate.
- A Mersenne prime is a prime of the form $2^n – 1$. Such a number is prime if and only if $2^{n-1}(2^n-1)$ is a perfect number (a number equal to the sum of all its proper divisors). All even perfect numbers arise this way, and no one knows if an odd perfect number exists.
- There are 51 known Mersenne primes, and it is unknown whether there are infinitely many.
- The largest prime that humans have ever found is $2^{82,589,933}-1$ and is a Mersenne prime. It was discovered in 2018.
- It is pretty easy to come up with a new kind of prime, such as a “palindromic prime”, but most of the time it is very difficult to prove that there are infinitely many such primes. In case you’re curious, 142840628019121910826048241 is a palindromic prime.
- If $a$ and $b$ are relatively prime nonzero integers, then there are infinitely many primes in the sequence $a + b, 2a + b, 3a+b,\dots$.
- The primes 17 and 37 are of the form $x^2+1$ for $x = 4$ and $x=6$ respectively. No one knows whether there are infinitely many primes of this form.
- If $F$ is a field with finitely many elements, then it must have $p^n$ elements for some prime and positive natural number $n$.
- Every positive natural number can be written uniquely (up to order of factors) as a product of prime numbers.
- Wilson’s theorem: If $n > 1$ is a natural number, then $(n-1)!\equiv -1\pmod{n}$ if and only if $n$ is prime. This theorem is totally impractical for primality testing in practice.
- There exists a real number $\theta$ such that $f(n) = \lfloor \theta^{3n}\rfloor$ is prime for every $n\geq 1$.
- Goldbach’s conjecture is that every even number $n\geq 4$ can be written as the sum of two primes. Indeed, experiments suggest that as $n$ increases, so does the number of ways it can be so written.

- …on the other hand, Ivan Matveevich Vinogradov proved that every sufficiently large
*odd*number can be written as the sum of three primes. - The infinite sum $1/2 + 1/3 + 1/5 + 1/7 + 1/11 + \cdots$ of the reciprocal of all primes diverges.
- Did I mention there are infinitely many primes? That is because if you only have finitely many primes $p_1,\dots,p_n$ then the number $p_1\cdots p_n + 1$ either is prime or has a prime factor not in the original list of primes. This proof is due to Euclid.
- It’s 2019. Unfortunately, that’s not a prime. The next prime year is 2027, and the one after that is 2029.
- You probably noticed that 2027 and 2029 are
*twin primes*. That is, they are prime numbers only two apart. There are probably infinitely many of them, but no one has proved this. - Even though no one has proved the twin prime conjecture, Yitang Zhang in 2013 proved that there are infinitely many prime pairs whose distance is 70,000,000 or less.
- Like most other well-studied types of primes, mathematicians have inevitably searched for huge examples of twin primes. For example, in 1994, Karl-Heinz Indlekofer and Antal Járai wrote a computer program that determined $697053813\times 2^{16352}\pm 1$ to be twin primes.
- The first prime after a trillion (10^12) is 1000000000039.
- Fermat’s little theorem states that if $p$ is a prime and $p$ does not divide an integer $a$ then $a^{p-1}\equiv 1\pmod{p}$. This can be used as a quick primality test. For example, $3^{1000} – 1$ is congruent to 990 modulo 1001, so 1001 is not prime (it is in fact divisible by 11).
- Fermat’s little theorem cannot always recognize composite numbers. In fact, there are numbers $n$ that are not prime, and yet $a^{n-1}\equiv -1\pmod{n}$ for every $a$. Such $n$ are called Carmichael numbers. Sneaky buggers.
- The smallest Carmichael number is 561, and Alford, Granville, and Pomerance proved in 1994 that there are infinitely many Carmichael numbers.
- Never fear, Fermat’s little theorem can be enhanced to better work against Carmichael numbers in what is known as the Miller-Rabin primality test.
- If $n\geq 5$ is prime then

\[ \binom{2n-1}{n-1}\equiv 1\pmod{n^3}. \] The converse is still a conjecture! - Remember Euclid’s proof that there are infinitely many primes? If $p$ is a prime and $N_p$ is the product of all primes less than or equal to $p$, then $N_p+1$ is either prime or has a prime factor larger than $p$. But when is $N_p$ actually prime? $p=24029$ is the largest prime known such that $N_p + 1$ is prime.
- Another type of prime is a Fermat prime. It is a prime of the form $F_n = 2^{2^n} + 1$. The largest known prime of this form is $F_4 = 65537$. So for example, $F_5 = 641\times 6700417$ and $F_6 = 274177\times 67280421310721$. It is unknown whether $F_{24}$ is prime or composite. The number $F_{24}$ is around $10^{5050445}$.
- Théophile Pépin gave a criterion for a Fermat number $F_n$ to be prime: it is prime if and only if $3^{(F_k-1)/2}\equiv -1\pmod{F_k}$.
- Lucas proved that if $a$ and $n \gt 1$ are integers and $a^{n-1}\equiv 1\pmod{n}$ but $a^{(n-1)/q}\not\equiv 1\pmod{n}$ for each prime $q\mid n-1$ then $n$ is prime.
- Every prime number of the form $4k + 1$ can be written as the sum of two squares. For example, $13 = 2^2 + 3^2$. No prime of the form $4k + 3$ can be so written.
- For every prime $p$ there exists integers $x$ and $y$ such that $x^2 + y^2 + 1$ is divisible by $p$.
- There is no nonconstant polynomial which takes on only prime values.
- However, the polynomial $f(x) = x^2 + x + 41$ is prime for $x=0,1,\dots, 39.$
- Stranger still, there exists a multivariable polynomial with integer coefficients whose values is exactly the set of primes when its variables take on all the nonnegative integers.
- Let $\pi(x)$ denote the number of primes less than or equal to $x$. The prime number theorem states that $\pi(x)$ is asymptotically $x/\log(x)$. That is, $\lim_{x\to\infty} \pi(x)/(x/\log(x))=1$.
- The current record for calculating $\pi(x)$ is

\[\pi(10^{25}) = 176846309399143769411680, \] found by Jan Büthe in 2014. - Gauss gave a more accurate estimate of ${\rm Li}(x):= \int_2^x {\rm d}t/\log(t)$ for $\pi(x)$. The function ${\rm Li}(x)$ is called the logarithmic integral.
- There are arbitrarily large sequences of consecutive integers, none of which are prime. One way to produce such a sequence is $n! + 2, n! +3,\dots, n! + n$, which is a sequence of $n-1$ composite numbers.
- The number 9012345678901 whose digits are consecutive integers, is a prime number! Here is another:

45678901234567890123456789012345678901234567890123

45678901234567890123456789012345678901234567890123

45678901234567890123456789012345678901234567890123

45678901234567890123456789012345678901234567Can you find a bigger one?!

- Legendre’s conjecture states that for every natural number $n$, there exists a prime number between $n^2$ and $(n+1)^2$.
- Although no one has proved Legendre’s conjecture, Chen Jingrun did prove that there exists a number $x$ between $n^2$ and $(n+1)^2$ that is either prime or a product of two primes.
- Chen also proved an approximation to Goldbach’s conjecture: every sufficiently large even number is the sum of a prime and a number that has at most two prime factors.
- Levy’s conjecture states that all odd numbers greater than or equal to 7 are the sum of a prime and twice a prime. For example: 11 = 5 + 2(3).
- A Sophie Germain prime is a prime $p$ such that $2p+1$ is also a prime. These primes are primes for which the “first case” of Fermat’s Last Theorem holds. No one knows if there are infinitely many.
- The largest Sophie Germain prime known to us is

\[ p = 2618163402417\times 2^{1290000} -1. \] - A
*Smarandache-Wellin*is a prime of the form $Q_n = p_1|p_2|\cdots |p_n$ where $p_1,\dots,p_n$ are the first $n$ primes and $a|b$ denotes $a$ concatenated with $b$. For example, $Q_1 = 2, Q_2 = 23, Q_3 =235$, etc. The first few running primes are $2, 23, 2357,\dots$ and the next one is quite large. I just made this definition up, but after searching the OEIS I found it it had the name Smarandache-Wellin. Pretty cool, right? Can you find the next one? - You can also stick a decimal point in front of the concatenation of primes to form $0.235711\dots$. This number is called the Copeland–Erdős constant and it’s irrational.
- There exists a polynomial-time algorithm to determine whether a number is prime. It was discovered in 2002 by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena.
- The Riemann zeta function is the analytic continuation of the function $\zeta(s) = 1^{-s} + 2^{-s} + 3^{-s} + \cdots$. It has trivial zeroes at the negative even integers. The Riemann hypothesis is that all its nontrivial zeroes are on the line Re=1/2, and is a Millenium prize problem worth a million USD. The Riemann hypothesis implies many facts about the distribution of primes, including the prime number theorem.