# NextPrime NextPrime[x]

gives the smallest prime above .

NextPrime[x,k]

gives the  -next prime above .

# Details • NextPrime is also known as previous prime.
• Mathematical function, suitable for both symbolic and numerical manipulation.
• For positive x, NextPrime[x] is the (m+1) prime number where m is the number of primes less than or equal to x.
• • NextPrime[x, k] is the (m+k) prime number where m is the number of primes less than or equal to x.
• # Examples

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## Basic Examples(2)

Compute the next prime after 10:

Plot the sequence of primes:

## Scope(9)

### Numerical Manipulation(7)

NextPrime works over integers:

Rational numbers:

Real numbers:

Compute the last prime before 50:

Compute the second prime after 50:

NextPrime works over large integers:

### Symbolic Manipulation(2)

Find a solution instance of equalities with NextPrime:

Reduce an equation:

## Applications(15)

### Basic Applications(6)

Highlight the sequence of primes using NextPrime:

Visualize the primes with a parabolic sieve:

Visualize the sieve of Eratosthenes for the first 5 primes:

Spiral of a NextPrime sequence:

Hexagonal prime spiral using NextPrime:

Find the lowest-divisor pair for an integer :

Find the prime factorization for an integer :

Construct a factorization tree for an integer :

### Approximations(2)

Approximation to Prime:

Plot Prime compared with an estimate:

Compute Prime:

### Number Theory(5)

NextPrime numbers modulo 10:

NextPrime numbers modulo :

Plot prime gaps around powers of 10:

Visualize the gaps between primes below 104:

Plot a histogram of the closest distances to a prime for a range of integers:

Compare the NextPrime sequence to PrimePi:

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

Verify this for a range of integers:

### Special Sequences(2)

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

Plot the twin primes:

Compute the smallest n-digit prime:

Plot the difference between the smallest n-digit prime and :

## Properties & Relations(4)

Primes represents the domain of all primes numbers:

The largest domain of definitions of NextPrime:

Compute the prime using NextPrime:

Compute using PrimePi:

## Neat Examples(5)

Ulam spiral of the distance to the next prime:

Generate a path based on the NextPrime sequence:

Visualize the increase in prime numbers through -sided polygons:

Plot NextPrime using ArrayPlot:

Sunflower seed representation of NextPrime: