Prime works over positive integers:

Large numbers:

Prime works over large integers:

Prime threads over lists:

TraditionalForm formatting:

Find a solution instance of equalities with Prime:

Sum involving prime numbers:

Product:

Recognize the Prime sequence:

Simplify an expression:

Find the leading asymptotic term for Prime as approaches Infinity:

The ratio of the sequence and its leading asymptotic approaches 1 as approaches Infinity:

Highlight prime numbers:

Visualize the primes with a parabolic sieve:

Visualize the sieve of Eratosthenes for the first 5 primes:

Spiral of primes:

Hexagonal prime spiral:

Find the lowest-divisor pair for an integer :

Find the factorization for an integer :

Construct a factorization tree for an integer :

Divisibility graph:

Visualize prime numbers with periodic curves. Prime numbers are those where only two curves cross—its own and the curve of 1:

Visualize an array of primes:

Approximation to with :

Plot Prime compared with an estimate:

Cesàro approximation of :

The relative error is small:

Compute Prime using the aliquot sum , which is equal to 1 when is prime:

Prime numbers modulo 10:

Prime numbers modulo :

Plot of the first 50 primes modulo 6:

Fraction of the first million primes greater than 3 of the form :

With the exception of 2 and 3, all primes are of the form :

Define a residue number system:

Choose two numbers and represent them in the residue system:

Multiplying and recovering in the residue system:

Adding and recovering:

An approximation to the zeta function:

Calculate the probability that a number is relatively prime to the first 25 primes:

Differences between the first 100 primes:

Find the jumping champion, i.e. the most frequently occurring difference between consecutive primes:

Calculate the sum of prime factors:

Plot the sum of prime factors for the first 25 integers:

Find Goldbach partitions, i.e. pairs of primes (, ) such that :

Graph Goldbach's conjecture/comet:

Calculate the primorial up to the n^thprime, i.e. a function that multiplies successive primes, similar to the factorial:

Compare the primorial to the factorial up to :

Plot the differences between the factorial and the primorial up to :

Plot the Chebyshev theta function:

Calculate the prime powers up to :

Count all the prime powers up to :

Graph the sequence of prime powers:

It is conjectured that for any integer , there is a prime with :

Verify this for a range of integers:

Plot Andrica's conjecture, which states that √p_n+1-√p_n<1 for every pair of consecutive prime numbers p_n and p_n+1:

Plot Brocard's conjecture, which states that if p_n and p_n+1 are consecutive primes greater than 2, then between (p_n)² and (p_n+1)² there are at least four prime numbers:

Plot polynomials for -gons, i.e. Chebyshev polynomials of the second kind, such that is prime:

Find twin primes, i.e. pairs of primes of the form :

Plot the twin primes:

A prime is a Sophie Germain prime if is also a prime:

Find Mersenne primes:

There are 5 Fermat primes, i.e. Fermat numbers such that is prime:

There are two Wieferich primes under , i.e. a prime number such that divides :

Construct prime Euclid numbers:

Find perfect numbers:

Pythagorean primes can be written as the sum of two squares, i.e. a prime number :

Ramanujan primes are the smallest number R_n such that π(x) - π(x/2) ≥ n for all x≥ R_n:

Differences in n^th Ramanujan primes and Prime:

Primes of the form , i.e. prime elements of the Gaussian integers:

Array plot of the Gaussian primes, a Gaussian integer such that if and are nonzero, then is prime or if , or if , :

Emirp primes are primes whose reversal is also prime but not a palindrome:

Find home primes of . is found by concatenating the prime factors of and repeating until a prime is reached: