# Prime

Prime[n]

gives the n prime number .

# Details • Prime is also known as prime number sequence.
• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• is the smallest positive integer greater than , which cannot be divided by any integer greater than 1 and smaller than itself. The first prime is the integer 2.
• is asymptotically equivalent to as .
• # Examples

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## Basic Examples(3)

Compute the prime:

Plot the sequence of primes:

Find the leading asymptotic term for Prime at Infinity:

## Scope(10)

### Numerical Manipulation(4)

Prime works over positive integers:

Large numbers:

Prime works over large integers:

### Symbolic Manipulation(6)

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:

## Applications(38)

### Basic Applications(9)

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 crossits own and the curve of 1:

Visualize an array of primes:

### Approximations(3)

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:

### Number Theory(14)

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:

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 nthprime, 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 pn+1-pn<1 for every pair of consecutive prime numbers pn and pn+1:

Plot Brocard's conjecture, which states that if pn and pn+1 are consecutive primes greater than 2, then between (pn)2 and (pn+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:

### Special Sequences(12)

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 Rn such that π(x) - π(x/2) n for all x Rn:

Differences in nth 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:

## Properties & Relations(8)

The traditional mathematical notation for Prime:

Primes represents the domain of all prime numbers:

The largest domain of definitions of Prime:

Prime is asymptotically equivalent to as :

PrimePi is the inverse of Prime:

Use NextPrime to find the next prime above n:

PrimeOmega counts the number of prime divisors:

EulerPhi[n] counts numbers less than or equal to that are coprime to :

## Possible Issues(1)

Evaluation timing increases exponentially:

## Interactive Examples(2)

Visualize factorization diagrams:

The polar plot of primes:

## Neat Examples(6)

Ulam spiral colored based on the difference in Prime values:

Visualize when is divisible by primes. Each row of dots corresponds to the divisors of , which are labelled along the horizontal axis:

Generate a path based on the Prime sequence:

Construct polyhedra using directed graphs generated by primes less than :

Visualize prime -sided polygons:

Fill the set of primes with primes: