NextPrime

The next prime strictly greater than n, or the kth prime after n.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPrime number↗MathWorldNextPrime↗PrimeNumber↗WikidataQ49008↗Encyclopedia of MathematicsPrime_number↗nLabprime number↗Britannicatopic/prime-number↗OEISA000040↗Wolfram LanguageNextPrime✓↗

NextPrime(n)smallest prime strictly greater than $n$.

SageMathnext_prime(n)SymPynextprime(n)Julianext_prime(ZZ(n))Rustnext_prime(n)

NextPrime(n, k)the $k$th prime after $n$; negative $k$ walks backward.

SymPynextprime(n, k)

NextPrime(n, k?)The next prime strictly greater than n, or the kth prime after n.

SymPynextprime(n, k?)

Typed signatures

Type	Package	Applies
(integer, integer?) -> integer	compute-engine	always
(number, number?) -> integer	number-theoryreplaces compute-engine	always

Domain: Number theory

Details

• NextPrime(n) is the $(m+1)$th prime, where $m$ is the count of primes $\le n$.
• NextPrime(n, k) generalizes to the $(m+k)$th prime, so a negative $k$ steps backward to a prime below $n$.
• n need not be prime or even an integer; compute-engine simply finds the next prime above it.
• Threads element-wise over a list, as Wolfram's Listable heads do.

Plot

Examples

1 more kept as data, checked against the oracles

Basic #

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In$\mathrm{NextPrime}(10)$Out$11$

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In$\mathrm{NextPrime}(100)$Out$101$

The second prime after 10

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In$\mathrm{NextPrime}(10, 2)$Out$13$

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In$\mathrm{NextPrime}(6)$Out$7$

The second prime after 50

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In$\mathrm{NextPrime}(50, 2)$Out$59$

The largest prime below 50

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In$\mathrm{NextPrime}(50, -1)$Out$47$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\mathrm{NextPrime}(\bigl\lbrack1, 5, 10, 15\bigr\rbrack)$Out$\bigl\lbrack2, 7, 11, 17\bigr\rbrack$

Large integers: the first prime past a googol is $10^{100}+267$

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In$\mathrm{NextPrime}(10^{100})$Out$10\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,000\,267$

Rational $n$: the next prime above $7/2$

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In$\mathrm{NextPrime}(\frac{7}{2})$Out$5$

Real $n$: the next prime above $100.5$

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In$\mathrm{NextPrime}(100.5)$Out$101$

Solves for $p$ whose next prime is $p+1$; only $p=1$ in range

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In$\mathrm{FindInstance}(\mathrm{NextPrime}(p)=p+1∧0\lt p\lt10, p, ℤ)$Out$\bigl\lbrack\bigl\lbrack\mathrm{Rule}(p, 1)\bigr\rbrack\bigr\rbrack$

Applications #

Second argument steps $k$ primes on from the start: the 25th prime

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In$\mathrm{NextPrime}(1, 25)$Out$97$

Last digits of the first 100 next-primes show the pattern of residues

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In$\mathrm{Array}(\mathrm{NextPrime}, 100)\bmod10$Out$\bigl\lbrack2, 3, 5, 5, 7, 7, 1, 1, 1, 1, 3, 3, 7, 7, 7, 7, 9, 9, 3, 3, 3, 3, 9, 9, 9, 9, 9, 9, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 1, 1, 3, 3, 7, 7, 7, 7, 3, 3, 3, 3, 3, 3, 9, 9, 9, 9, 9, 9, 1, 1, 7, 7, 7, 7, 7, 7, 1, 1, 1, 1, 3, 3, 9, 9, 9, 9, 9, 9, 3, 3, 3, 3, 9, 9, 9, 9, 9, 9, 7, 7, 7, 7, 7, 7, 7, 7, 1, 1, 1, 1\bigr\rbrack$

Lesser members of twin prime pairs below 100

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In$\mathrm{Filter}(1..100, \operatorname{\_1}↦\mathrm{IsPrime}(\operatorname{\_1})∧\mathrm{NextPrime}(\operatorname{\_1})=\operatorname{\_1}+2)$Out$\bigl\lbrack3, 5, 11, 17, 29, 41, 59, 71\bigr\rbrack$

First prime above each power of ten

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In$\mathrm{Map}(\operatorname{\_1}↦\mathrm{NextPrime}(10^{\operatorname{\_1}}), 0..20)$Out$\bigl\lbrack2, 11, 101, 1\,009, 10\,007, 100\,003, 1\,000\,003, 10\,000\,019, 100\,000\,007, 1\,000\,000\,007, 10\,000\,000\,019, 100\,000\,000\,003, 1\,000\,000\,000\,039, 10\,000\,000\,000\,037, 100\,000\,000\,000\,031, 1\,000\,000\,000\,000\,037, 10\,000\,000\,000\,000\,061, 100\,000\,000\,000\,000\,003, 1\,000\,000\,000\,000\,000\,003, 10\,000\,000\,000\,000\,000\,051, 100\,000\,000\,000\,000\,000\,039\bigr\rbrack$

Gap between $10^n$ and the next prime stays small

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In$\mathrm{Map}(\operatorname{\_1}↦\mathrm{NextPrime}(10^{\operatorname{\_1}})-10^{\operatorname{\_1}}, 0..20)$Out$\bigl\lbrack1, 1, 1, 9, 7, 3, 3, 19, 7, 7, 19, 3, 39, 37, 31, 37, 61, 3, 3, 51, 39\bigr\rbrack$

Properties #

NextPrime steps from $p_n$ to $p_{n+1}$. See NthPrime

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In$\mathrm{NextPrime}(\mathrm{NthPrime}(20))=\mathrm{NthPrime}(21)$Out$\mathrm{True}$

The 39th prime, reached by stepping from 1

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In$\mathrm{NextPrime}(1, 39)$Out$167$

Skips the composite 16 to land on 17

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In$\mathrm{NextPrime}(15)$Out$17$

The result is always a member of the primes

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In$\mathrm{NextPrime}(30)∈\mathbb{P}$Out$\mathrm{True}$

Possible issues #

A negative $k$ walks backward, giving the prime immediately before 10

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In$\mathrm{NextPrime}(10, -1)$Out$7$

Negative $n$: Wolfram counts $-7$ as prime and returns it

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In$\mathrm{NextPrime}(-10)$Out$-7$

About

Examples	23
Checked against	Wolfram Language 23 agreeSageMath 8 agree1 differ2 errorSymPy 14 agree1 differ2 errorJulia 7 agree1 differ2 errorRust 6 agree5 error
Crosswalk	WikipediaMathWorld ×2WikidataEncyclopedia of MathematicsnLabBritannicaOEISWolfram LanguageSageMathSymPy ×2JuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

nextprime(n) is the least prime above n.

mappedsympypackages/oracle/src/mappings.ts

nextprime(n, i) is the i-th prime above n; i must be positive, where ours takes a negative step to the primes below.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: NthPrime, PrimePi