What is Prime Number?

Use this calculator to compute Prime Number values with step-by-step solutions.

How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Prime Number used?

Arithmetic, calculus, physics, engineering, and related disciplines.

What formulas does this use?

All standard prime number formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

5 more

NUMBER THEORYArithmeticMathematics Calculator

🔢

Prime Numbers: Only 1 and Itself

A prime has exactly two divisors: 1 and itself. 2,3,5,7,11,... Sieve of Eratosthenes finds primes up to N. Twin primes differ by 2.

Concept Fundamentals

2 divisors

Prime

First primes

2,3,5,7

O(N log log N)

Sieve

p, p+2

Twin

Did our AI summary help? Let us know.

Prime: only divisors 1 and itself. 2 is the only even prime. Sieve of Eratosthenes: O(N log log N) to find primes ≤ N. Twin primes: (3,5), (11,13). Conjecture: infinitely many.

Key quantities

2 divisors

Prime

Key relation

First primes

2,3,5,7

Key relation

O(N log log N)

Sieve

Key relation

p, p+2

Twin

Key relation

Ready to run the numbers?

Why: Primes are the building blocks of integers. RSA cryptography relies on large primes. Infinitely many primes (Euclid). Twin prime conjecture.

How: Trial division: check divisors up to √n. Sieve of Eratosthenes: mark multiples of 2,3,5,... Nth prime: sieve or approximate n ln n.

Prime: only divisors 1 and itself. 2 is the only even prime.Sieve of Eratosthenes: O(N log log N) to find primes ≤ N.

Sources:Khan AcademyOEIS A000040

Run the calculator when you are ready.

Check PrimalityEnter number or nth

Enter Number to check

Number

prime.sh

CALCULATED

$ isprime 31

Result

Prime ✓

Twin primes nearby

(11,13), (17,19), (29,31), (41,43)

Prime Number Calculator

31 is Prime

numbervibe.com

Result

📐 Step-by-Step Breakdown

SETUP

RESULT

Is prime?

Yes

Check up to √n

√31 ≈ 5.57

ext{Only} ext{need} ext{to} ext{test} divisors leq √n

Twin primes nearby

(11,13), (17,19), (29,31)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🔢

Prime: only 1 and itself divide it

— Definition

📐

Sieve of Eratosthenes: O(N log log N)

— Algorithm

📋 Key Takeaways

• Prime: natural number > 1 with exactly two divisors: 1 and itself
• Primality test: check divisibility by primes from 2 to √n
• Sieve of Eratosthenes: find all primes up to N in O(N log log N)
• Twin primes: pairs (p, p+2) like (3,5), (11,13) — conjecture: infinitely many
• 2 is the only even prime — all others are odd

💡 Did You Know?

🔢2 is the only even prime. All other primes are odd.Source: Number Theory

📐Euclid proved there are infinitely many primes around 300 BCE.Source: Euclid's Elements

🔄Twin prime conjecture: infinitely many pairs (p, p+2)? Still unproven.Source: Open Problem

📊RSA cryptography uses the product of two large primes — factoring is hard.Source: Cryptography

📚Sieve of Eratosthenes (c. 240 BCE): oldest known algorithm for finding primes.Source: History

💡Prime number theorem: ~n/ln(n) primes ≤ n. 10th prime ≈ 29.Source: Analytic Number Theory

📖 How It Works

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. To check if n is prime: test divisibility by 2, 3, 5, 7, ... up to √n. If none divide n, n is prime. The Sieve of Eratosthenes marks multiples of each prime to find all primes up to a limit. The nth prime is found by counting primes from 2 upward.

📝 Worked Example: Is 31 Prime?

√31 ≈ 5.6. Test divisors: 2, 3, 5.

31 ÷ 2 = 15.5 (not integer). 31 ÷ 3 = 10.33. 31 ÷ 5 = 6.2.

None divide 31. 31 is prime.

🚀 Real-World Applications

🔐 Cryptography

RSA, Diffie-Hellman: security relies on hardness of factoring.

💻 Hashing

Prime table sizes reduce collisions in hash tables.

🎲 Random Numbers

Primes used in pseudo-random number generators.

📡 Error Correction

Reed-Solomon codes use finite fields of prime order.

🔬 Science

Prime periods in cicada life cycles (13, 17 years).

📊 Data Structures

Bloom filters, perfect hashing use primes.

⚠️ Common Mistakes to Avoid

1 is not prime: By definition, primes must be > 1. 1 has only one divisor.
Testing past √n: If no divisor ≤ √n divides n, n is prime. No need to test further.
Even numbers: Only 2 is prime. All other evens are divisible by 2.
Negative numbers: Primes are defined for positive integers only.
Confusing prime with coprime: Prime = number with 2 divisors. Coprime = gcd(a,b)=1.

🎯 Expert Tips

💡 6k±1 Test

All primes > 3 are of form 6k−1 or 6k+1. Test only those.

💡 Sieve

For "all primes up to N", sieve is O(N log log N) — very fast.

💡 Quick Checks

Even (except 2)? Composite. Sum of digits divisible by 3? Composite.

💡 Nth Prime

p_n ≈ n ln(n). 10th prime ≈ 10×ln(10) ≈ 23; actual 29.

📊 Reference Table

n	nth Prime	Note
1	2	Smallest
2	3
5	11
10	29
25	97
100	541

📐 Quick Reference

Only even prime

√n

Test up to this

∞

Infinitely many

6k±1

Form for p>3

🎓 Practice Problems

Is 97 prime? → Yes

Is 91 prime? → No (7×13)

10th prime? → 29

Twin primes < 20? → (3,5), (5,7), (11,13), (17,19)

❓ FAQ

What is a prime number?

A natural number greater than 1 with exactly two divisors: 1 and itself.

How to check primality?

Test divisibility by primes from 2 to √n. If none divide n, n is prime.

Is 1 prime?

No. By definition, primes must be > 1. 1 has only one divisor.

Why only check up to √n?

If n = a×b and a ≤ b, then a ≤ √n. So a factor exists only if some divisor ≤ √n divides n.

How many primes are there?

Infinitely many. Euclid's proof: assume finitely many, multiply and add 1 — not divisible by any prime.

What are twin primes?

Pairs (p, p+2) both prime: (3,5), (11,13). Conjecture: infinitely many.

What is the largest known prime?

As of 2024, 2^82,589,933 − 1 (Mersenne prime). New ones are found regularly.

📌 Summary

Primes are the building blocks of integers. Check by testing divisors up to √n. The Sieve of Eratosthenes finds all primes up to N efficiently. Primes are essential in cryptography, hashing, and number theory.

✅ Verification Tip

For composite: find a divisor. For prime: confirm no divisor ≤ √n. For nth prime: verify it's the nth in sequence (2,3,5,7,11,...).

🔗 Next Steps

Explore the Prime Factorization Calculator to decompose numbers into primes. The GCF Calculator uses primes for efficient computation.

⚠️ Disclaimer: For very large n, primality testing may be slow. Nth prime limited to n ≤ 1000.

👈 START HERE

⬅️Jump in and explore the concept!

Prime Numbers: Only 1 and Itself

Enter Number to check

Primes Nearby

Result

📐 Step-by-Step Breakdown

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How It Works

📝 Worked Example: Is 31 Prime?

🚀 Real-World Applications

🔐 Cryptography

💻 Hashing

🎲 Random Numbers

📡 Error Correction

🔬 Science

📊 Data Structures

⚠️ Common Mistakes to Avoid

🎯 Expert Tips

💡 6k±1 Test

💡 Sieve

💡 Quick Checks

💡 Nth Prime

📊 Reference Table

📐 Quick Reference

🎓 Practice Problems

❓ FAQ

What is a prime number?

How to check primality?

Is 1 prime?

Why only check up to √n?

How many primes are there?

What are twin primes?

What is the largest known prime?

📌 Summary

✅ Verification Tip

🔗 Next Steps

Related Mathematics Calculators

Related Calculators

Absolute Change Calculator

Absolute Value Calculator

Adding & Subtracting Fractions Calculator

Addition Calculator

Arithmetic Mean Calculator

Arithmetic Sequence Calculator

We Value Your Privacy