2026-09-29 8 min

Prime Number Calculator

Find the nth prime, prime-count π(x), and neighboring primes with a sieve — positions 1 to 10,000.

Calculator

Prime Position Inputs

Position (k) Help

Which prime to find: 1 to 10,000

Results update live as you type

Status:p(25) = 97π(97) = 25

Prime #25 Summary

kth prime97

Previous prime89

Next prime101

Forward gap4

Sieve ceiling113

Results

Optimal

Prime #25 — p(k)

The 25th prime

π(p) — Prime Count

Exactly 25 primes ≤ 97

Previous Prime

Largest prime below p(k)

Next Prime

101

Smallest prime above p(k)

Forward Prime Gap

next − p(k)

Sieve Ceiling Used

113

k·(ln k + ln ln k) bound

Eratosthenes sieving to a proven Rosser bound makes every indexed prime exact; π(p) = k restates the index as a self-check.

Visualizations

Neighborhood sizes

89prev

97p(k)

101next

Prime staircase to p(k)

Position in cap

k / 10,00025

Gap vs prime size

gap4

Formula

Crossing out multiples leaves exactly the primes in order; position k names the prime and the prime count simultaneously.

p(k) = kth survivor of the sieve; π(p(k)) = k

Alternative Form

Sieve ceiling: k·(ln k + ln ln k) + 3 (k ≥ 6)

Variables

Symbol	Name	Unit	Description
k	Position	count	1-based index into the prime list
p(k)	kth prime	integer	The prime at position k
π(x)	Prime counting	count	Number of primes ≤ x
gap	Prime gap	integer	next prime minus p(k)

Explanation

Sieving to a proven ceiling guarantees position k exists in the table; neighbors and gaps then come from short trial-division walks.

Derivation

Rosser: p(k) < k(ln k + ln ln k) for k ≥ 6. Sieve [2, bound], index survivors, verify count ≥ k, then walk ±1 with trial division for neighbors.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Read the requested position k.

Inputs: k

k:25

Result: Finding prime #25

Step 2

Size the sieve with the Rosser bound.

bound = k·(ln k + ln ln k) + 3

k:25

bound:113

Intermediate: ln 25 ≈ 3.219

Result: Sieve ceiling = 113

Step 3

Run Eratosthenes: cross out multiples of each prime.

Strike 2, then 3, then 5, …

ceiling:113

Intermediate: 25 survivors needed

Result: 113 sieved, position 25 secured

Step 4

Read off the kth survivor.

p(k) = kth uncrossed number

k:25

Result: p(25) = 97

Step 5

Step down and up with trial division for neighbors.

Test p−1, p−2, … and p+1, p+2, …

p:97

Intermediate: prev 89, next 101

Result: Neighbors: 89 ← 97 → 101

Step 6

Record the gap and the π(p) = k identity.

gap = next − p; π(p) = k

next:101

p:97

Result: Gap = 4; π(97) = 25

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

First double-digit positions.

Inputs

k:10

Calculation

Primes: 2,3,5,7,11,13,17,19,23,29

Result

p(10) = 29

Interpretation

Count ten primes starting at 2 to land on 29.

Understanding Your Result

Your Result: p(k), its neighbors, and the prime gap for your position — see the result cards.

Typical range: p(1) = 2 through p(10,000) = 104,729; gaps stay small (usually under 40) across this range.

What This Means

The kth prime is the prime sitting at index k; π(p) = k restates the same fact as a counting identity.

Why It Matters

Prime-indexed sizing (hash tables, sampling strides, cipher parameters) starts by naming primes by position.

Professional Interpretation

A gap of 2 flags twin primes; gaps above 30 this low are mildly unusual and worth double-checking by listing.

Recommendations

•Quote primes with their index, e.g. p(25) = 97
•Use π(p) = k as a self-check
•Read gap 2 as a twin pair
•Size hash tables at nearby primes
•Compare neighbors before assuming isolation

Educational Content

Introduction

Primes arrive in a fixed order, so each one has an address: its index k. The sieve writes the street map; indexing reads the house.

Concept Explanation

Cross out multiples of 2, then 3, then 5 — survivors are primes in order. A proven ceiling guarantees the kth survivor is already on the map.

Applications

▸Prime-sized hash tables

▸Deterministic sampling strides

▸Cipher parameter selection

▸Twin-prime exploration

▸Prime-gap statistics

▸Contest prime queries

Benefits

•Exact sieve, not estimation
•π(x) identity included
•Neighbors and gap free
•Bound shown for transparency
•Twin pairs flagged

Limitations

•Caps at k = 10,000
•Sieve memory grows with bound
•Estimates like k·ln k not reported

Best Practices

•State k before computing
•Verify with π(p) = k
•Check neighbors for twins
•Round table sizes to nearby primes
•Keep k within the cap

Common Misconceptions

Myth: The nth prime has a simple formula.

Truth: No closed form exists; sieves and estimates (k·ln k) are the practical tools.

Myth: π(x) means 3.14159… here.

Truth: π(x) is prime-counting notation — primes ≤ x — unrelated to the circle constant.

Myth: Prime gaps grow fast.

Truth: Below 105k, gaps average ~12; gaps above 100 first appear past 500.

Myth: 1 is the first prime.

Truth: The prime list starts at 2; p(1) = 2.

Practical Tips

•Quote the index with the prime: p(25) = 97
•Self-check with π(p) = k
•Read a gap of 2 as twin primes
•Round hash sizes to the prime below your target
•Compare with neighbors before citing isolation
•Use the bound to see sieve cost
•Keep k between 1 and 10,000

Common Mistakes

Starting the prime list at 1

1 is not prime; the list — and every index — starts at 2.

Confusing π(x) with pi the constant

Prime-counting π(x) counts primes ≤ x; notation overlaps, meaning does not.

Using k·ln k as exact

That estimates p(k); the sieve gives the exact prime — verify before citing.

Assuming gaps are always even

All gaps after (2,3) are even, but the 3−2 = 1 gap is the famous exception.

Requesting k beyond the cap

Positions stop at 10,000; larger queries need segmented sieves offline.

Assumptions

This calculator makes the following assumptions to provide estimates:

1-based indexing: p(1) = 2; k must be a positive integer.

Rosser bound: Ceiling k·(ln k + ln ln k)+3 holds for k ≥ 6; small k use 15.

Exact sieve: Eratosthenes marks composites deterministically.

Trial-division neighbors: Walks of ±1 use exact 6k±1 testing.

Local compute: Sieving runs in the browser; nothing is uploaded.

Limitations

Be aware of these limitations when using this calculator:

k ≤ 10,000: The 10,000th prime (104,729) sets the ceiling.

Sieve memory: The byte table grows with the bound; huge k needs segmentation.

No primality proofs shown: Neighbors use trial division, not certificates.

Single position: One k per run; ranges need repeated queries.

Frequently Asked Questions

Answers to common questions about this calculator.

What is the nth prime?

The prime at position n in 2, 3, 5, 7, 11, … — so the 1st prime is 2, the 10th is 29, and the 25th is 97.

How is the nth prime found here?

A Sieve of Eratosthenes crosses out multiples up to a proven upper bound k·(ln k + ln ln k), then reads off position k.

What is π(x), the prime-counting function?

π(x) counts primes ≤ x. Since p is exactly the kth prime, π(p) = k — the tool reports this identity for every result.

Why is k capped at 10,000?

The 10,000th prime is 104,729, needing a ~110k sieve — instant. Larger k would push sieves into the millions and stall the page.

How are the neighbor primes found?

Trial division steps down from p−1 for the previous prime and up from p+1 for the next — gaps near 100k are tiny, so this is instant.

What is a prime gap?

The difference between consecutive primes. The tool shows next − p; the first gap above 100 appears after the prime 523.

Are there infinitely many primes?

Yes — Euclid proved it around 300 BCE: multiply all known primes, add 1, and a new prime factor must appear.

What are twin primes?

Prime pairs differing by 2, like (29, 31). When the gap reads 2, you have found one.

How do I estimate the millionth prime without sieving?

The prime number theorem says p(k) ≈ k·ln k. For k = 10⁶ that gives ~13.8M (true answer 15,485,863) — close but not exact.

Does this tool store my queries?

No. Sieving runs locally in your browser.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Sieving by hand to 100

Cross-out order that never misses.

Reading π(x) tables

Counting primes without listing.

Twin-prime spotting

Gap-2 pairs near your primes.

Prime-sized hash tables

Why 997 beats 1,000 slots.

Glossary

Technical terms used in mortgage calculations explained in plain language.

nth prime p(k): The prime at index k of the ordered prime list.
Prime-counting π(x): How many primes are ≤ x.
Sieve of Eratosthenes: Crossing out multiples to leave primes.
Prime gap: Difference between consecutive primes.
Twin primes: Prime pair differing by 2.
Rosser bound: Proven ceiling k·(ln k + ln ln k) for p(k).
Trial division: Testing divisors ascending from 2.
Prime number theorem: π(x) ≈ x/ln x; p(k) ≈ k·ln k asymptotically.

Authoritative References

Trusted sources and standards referenced by this calculator.

Wikipedia — Prime number Wikipedia — Prime-counting function Wikipedia — Sieve of Eratosthenes Wikipedia — Prime gap MathWorld — Prime Number The Prime Pages (UTM) Khan Academy — Prime numbers

Applicable Standards

Common Core 4.OA.B.4 — prime and composite numbersNCTM Number and Operations — properties and representations

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Page Information

Last Updated:

2026-09-29

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.