NthPrime

The nth prime number (NthPrime(1) = 2).

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

NthPrime(n)the $n$th prime, with $\mathrm{NthPrime}(1)=2$.

SageMathnth_prime(n)✓↗SymPyprime(n)↗Rustnth_prime(n)

Typed signatures

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

Domain: Number theory

Details

• Inverse of PrimePi: $\pi(p_n)=n$.
• Grows asymptotically like $n\ln n$, by the prime number theorem.
• n must be a positive integer; compute-engine leaves non-positive n, including NthPrime(0), unevaluated.
• Threads element-wise over a list, as Wolfram's Listable heads do.

Plot

Examples

Basic #

The first prime is 2

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In$\mathrm{NthPrime}(1)$Out$2$

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In$\mathrm{NthPrime}(10)$Out$29$

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In$\mathrm{NthPrime}(100)$Out$541$

Scope #

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

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In$\mathrm{NthPrime}(\bigl\lbrack1, 3, 4, 10\bigr\rbrack)$Out$\bigl\lbrack2, 5, 7, 29\bigr\rbrack$

The millionth prime

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In$\mathrm{NthPrime}(1\,000\,000)$Out$15\,485\,863$

Applications #

A prime gap: $p_{31}-p_{30}=127-113$

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In$\mathrm{NthPrime}(31)-\mathrm{NthPrime}(30)$Out$14$

The 25th prime is the last one below 100

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In$\mathrm{NthPrime}(25)$Out$97$

Last digits of the first 100 primes; only 1, 3, 7, 9 after the first two

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

Mertens' product $\prod(1-1/p)$ over 25 primes, near $e^{-\gamma}/\ln p$

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In$N(100\mathrm{Last}(\mathrm{FoldList}(\mathrm{Multiply}, 1, 1-\frac{1}{\mathrm{NthPrime}(1..25)})))$Out$12.031\,729\,047\,493\,518\,845\,046\,377\,749\,916\,002\,447\,517\,547\,079\,907\,308\,611\,992\,957\,558\,959\,774\,039\,467\,813\,931\,634\,754\,337\,362\,993\,103\,663\,911\,632\,323\,741\,142\,560\,525\,368\,787\,535\,602\,215\,437\,513\,280\,363\,740\,373\,267\,063\,423\,780\,504\,807\,034\,048\,741\,363\,892\,815\,775\,512\,794\,985\,020\,245\,119\,743\,682\,156\,856\,905\,745\,149\,365\,745\,047\,058\,318\,405\,826\,205\,440\,410\,170\,440\,671\,667\,429\,458\,821\,773\,900\,960\,139\,332\,334\,824\,297\,814\,727\,606\,130\,865\,211\,851\,808\,520\,213\,524\,696\,348\,571\,409\,991\,875\,632\,191\,069\,982\,658\,715\,093\,864\,363\,704\,286\,287\,076\,786\,697\,026\,702\,666\,817\,476\,899\,380\,475\,305\,246\,364\,654\,764\,856\,705\,150\,394\,009\,025\,529\,594\,469\,766\,357\,381\,674\,148\,101\,642\,444\,8$

The fourth prime, 7, is a Mersenne prime exponent

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In$\mathrm{IsMersennePrimeExponent}(\mathrm{NthPrime}(4))$Out$\mathrm{True}$

$2^7-1=127$ is prime

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In$\mathrm{IsPrime}(2^{\mathrm{NthPrime}(4)}-1)$Out$\mathrm{True}$

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In$\frac{1}{\prod_{n=1}^6(1-\mathrm{NthPrime}(n)^{-s})}$Out$\frac{1}{(1-2^{-s})(1-3^{-s})(1-5^{-s})(1-7^{-s})(1-11^{-s})(1-13^{-s})}$

Properties #

PrimePi and NthPrime are inverses: $\pi(p_n)=n$

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In$\mathrm{PrimePi}(\mathrm{NthPrime}(100))=100$Out$\mathrm{True}$

PrimePi inverts NthPrime

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In$\mathrm{PrimePi}(\mathrm{NthPrime}(100))$Out$100$

The next prime after 30 is $p_{\pi(30)+1}$

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In$\mathrm{NthPrime}(1+\mathrm{PrimePi}(30))$Out$31$

A prime has exactly one prime factor

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In$\mathrm{PrimeOmega}(\mathrm{NthPrime}(5))$Out$1$

A prime's only divisors are 1 and itself

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In$\mathrm{Divisors}(\mathrm{NthPrime}(5))$Out$\bigl\lbrack1, 11\bigr\rbrack$

$\varphi(p)=p-1$

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In$\mathrm{Totient}(\mathrm{NthPrime}(10))$Out$28$

The only even prime is $p_1=2$, so $p_2$ is the smallest odd prime

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In$\mathrm{NthPrime}(2)$Out$3$

Known value · Wolfram MathematicalFunctionData, Prime

Primes begin 2, 3, 5

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In$\mathrm{NthPrime}(3)$Out$5$

Known value · Wolfram MathematicalFunctionData, Prime

The fourth prime

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In$\mathrm{NthPrime}(4)$Out$7$

Known value · Wolfram MathematicalFunctionData, Prime

The fifth prime

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In$\mathrm{NthPrime}(5)$Out$11$

Known value · Wolfram MathematicalFunctionData, Prime

The sixth prime

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In$\mathrm{NthPrime}(6)$Out$13$

Known value · Wolfram MathematicalFunctionData, Prime

Seventh prime, after a gap of 4

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In$\mathrm{NthPrime}(7)$Out$17$

Known value · Wolfram MathematicalFunctionData, Prime

Output is always a member of the primes

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

One less than $p_{10}=29$ equals $\varphi(29)$

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In$\mathrm{NthPrime}(10)-1$Out$28$

The eighth prime, a twin with 17

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In$\mathrm{NthPrime}(8)$Out$19$

Known value · Wolfram MathematicalFunctionData, Prime

The ninth prime

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In$\mathrm{NthPrime}(9)$Out$23$

Known value · Wolfram MathematicalFunctionData, Prime

Starts the twenties-to-thirties stretch

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In$\mathrm{NthPrime}(11)$Out$31$

Known value · Wolfram MathematicalFunctionData, Prime

The twelfth prime

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In$\mathrm{NthPrime}(12)$Out$37$

Known value · Wolfram MathematicalFunctionData, Prime

The thirteenth prime

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In$\mathrm{NthPrime}(13)$Out$41$

Known value · Wolfram MathematicalFunctionData, Prime

Twin to 41

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In$\mathrm{NthPrime}(14)$Out$43$

Known value · Wolfram MathematicalFunctionData, Prime

The fifteenth prime

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In$\mathrm{NthPrime}(15)$Out$47$

Known value · Wolfram MathematicalFunctionData, Prime

The sixteenth prime

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In$\mathrm{NthPrime}(16)$Out$53$

Known value · Wolfram MathematicalFunctionData, Prime

Twin to 61

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In$\mathrm{NthPrime}(17)$Out$59$

Known value · Wolfram MathematicalFunctionData, Prime

The eighteenth prime

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In$\mathrm{NthPrime}(18)$Out$61$

Known value · Wolfram MathematicalFunctionData, Prime

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In$\sin(\frac{1}{\mathrm{NthPrime}(2)}π(1+Γ(\mathrm{NthPrime}(2))))^2+\sin(π\mathrm{NthPrime}(2))^2$Out$0$

Known value · Wolfram MathematicalFunctionData, Prime, FunctionalEquations

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In$\frac{e^{(2i πΓ(\mathrm{NthPrime}(2)))/\mathrm{NthPrime}(2)}-1}{e^{(-2i π)/\mathrm{NthPrime}(2)}-1}$Out$1$

Known value · Wolfram MathematicalFunctionData, Prime, FunctionalEquations

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In$\sin(\frac{1}{\mathrm{NthPrime}(3)}π(1+Γ(\mathrm{NthPrime}(3))))^2+\sin(π\mathrm{NthPrime}(3))^2$Out$0$

Known value · Wolfram MathematicalFunctionData, Prime, FunctionalEquations

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In$\frac{e^{(2i πΓ(\mathrm{NthPrime}(3)))/\mathrm{NthPrime}(3)}-1}{e^{(-2i π)/\mathrm{NthPrime}(3)}-1}$Out$1$

Known value · Wolfram MathematicalFunctionData, Prime, FunctionalEquations

Possible issues #

n must be a positive integer; compute-engine leaves NthPrime(0) unevaluated

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In$\mathrm{NthPrime}(0)$Out$\mathrm{NthPrime}(0)$

Neat examples #

The 1000th prime

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In$\mathrm{NthPrime}(1\,000)$Out$7\,919$

About

Examples	42
Checked against	Wolfram Language 42 agreeSageMath 36 agree2 errorSymPy 36 agree1 errorRust 30 agree2 differ1 error
Crosswalk	WikipediaMathWorldWikidataEncyclopedia of MathematicsnLabBritannicaOEISWolfram LanguageSageMathSymPyRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

prime(n) is SymPy's n-th prime.

mappedsagepackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

See also: FactorInteger, PrimePi