Prime number

A prime number is an integer $p>1$ whose only positive divisors are $1$ and $p$.

For every integer $n>1$, there is a prime number strictly between $n$ and $2n$. This supplies cyclic prime moduli comparable to an integer interval's length in additive combinatorics.

Two primes differing by two form a twin-prime pair, for example $3,5$ or $11,13$. The twin-prime upper bound using Bombieri–Vinogradov limits their number without proving there are infinitely many.

Apply the Selberg upper-bound sieve to the numbers $p+2$ with $p\le x$ prime. For odd squarefree integers $d$, the local count is $\pi (x;d,-2)$ and the density is $1/\varphi (d)$. With sieve level $D=x^{1/3}$, the normalizing sum is $\gg \log x$ by the truncated Euler-product lower bound. The Selberg least-common-multiple weights have absolute values at most $3^{\omega (d)}$. The weighted arithmetic-progression error bound and Bombieri–Vinogradov theorem make their total error $O(x/{\log }^{3}x)$. Consequently the number of twin prime pairs with smaller member at most $x$ is $O(x/{\log }^{2}x)$.

There are infinitely many prime numbers. Indeed, for any finite list of prime numbers $p_1,\dots ,p_k$, the integer $N=p_1\cdots p_k+1>1$ has a prime factor, by the Fundamental theorem of arithmetic. None of the listed prime numbers divides $N$, since each divides $N-1$ and a common divisor would divide $1$. Thus no finite list contains all prime numbers. The constructed $N$ need not itself be a prime number.

An odd prime is a prime number that is an odd integer. The only even number that is a prime number is $2$, since every other positive even number has $2$ as a proper divisor. Thus every prime number greater than $2$ is an odd prime. There are infinitely many odd primes, by Euclid's theorem.

A Fermat prime is a prime number of the formFor $k\ge 1$, it is congruent to $1$ modulo $4$ and to $2$ modulo $3$.

For a Fermat prime $p=2^{2^k}+1$, the multiplicative group modulo $p$ has order $2^{2^k}$. Its $\phi (p-1)=(p-1)/2$ primitive roots are all quadratic nonresidues; since there are also $(p-1)/2$ nonresidues, the two sets coincide.

Every integer greater than one is a product of prime numbers, uniquely up to reordering the factors.

A prime factor of a nonzero integer is a prime number that divides it.

A prime power is a positive integer of the form $p^d$, where $p$ is a prime number and $d$ is a nonnegative integer.