Is 37 a Prime Number?

What this calculator does

Enter a number and get three things at once: whether it is prime, a complete list of its divisors, and optionally every prime number from 2 up to your input. No formulas to remember - just type and read.

Quick start

Type 97 and click "Check primality". Result: YES - 97 is prime (divisors: 1, 97 only). Check "Find all primes in range" to see all 25 primes from 2 to 97.

What is a prime number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The number 6 is not prime because 6 = 2 x 3. The number 7 is prime because nothing divides it evenly except 1 and 7.

The fundamental theorem of arithmetic states that every integer greater than 1 can be represented uniquely as a product of prime numbers. Primes are the building blocks of all natural numbers.

First 50 prime numbers

Position	Prime	Position	Prime	Position	Prime
1st	2	11th	31	21st	73
2nd	3	12th	37	22nd	79
3rd	5	13th	41	23rd	83
4th	7	14th	43	24th	89
5th	11	15th	47	25th	97
6th	13	16th	53	26th	101
7th	17	17th	59	27th	103
8th	19	18th	61	28th	107
9th	23	19th	67	29th	109
10th	29	20th	71	30th	113

Practical examples

Is 97 prime? Enter 97. Result: YES. Divisors: 1, 97. Only two divisors means prime. There are 25 primes between 2 and 97.
97 is the 25th prime number

Is 100 prime? Enter 100. Result: NO. Divisors: 1, 2, 4, 5, 10, 20, 25, 50, 100. Nine divisors - highly composite. 100 = 2 x 2 x 5 x 5.
100 has 9 divisors - far from prime

Is 2 prime? Enter 2. Result: YES. The smallest and only even prime number. Every other even number is divisible by 2, so none can be prime.
2 is the only even prime - a unique case

Is 561 prime? Enter 561. Result: NO. Divisors: 1, 3, 11, 17, 33, 51, 187, 561. This is a Carmichael number - it passes some primality tests but is actually composite (3 x 11 x 17).
561 is the smallest Carmichael number - a "pseudoprime"

Primes up to 50: Enter 50 with "Find all primes" checked. Result: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 - that is 15 primes in the range 2-50.
About 30% of numbers under 50 are prime

How prime checking works

The calculator uses trial division: it checks whether any integer from 2 to the square root of n divides n evenly. If none does, n is prime. Why only up to the square root? Because if n = a x b and both a and b are greater than the square root of n, then a x b > n - a contradiction.

Method	How it works	Speed	Used for
Trial division	Divide by every number up to the square root	Slow	Small numbers, education
Sieve of Eratosthenes	Mark multiples of each prime as composite	Fast for ranges	Finding all primes up to N
Miller-Rabin	Probabilistic test using modular exponentiation	Very fast	Cryptography (RSA key generation)
AKS	Deterministic polynomial-time test	Theoretical	Proof that PRIMES is in P (2002)

Why primes matter

Field	How primes are used
Cryptography (RSA)	RSA encryption relies on the fact that multiplying two large primes is easy, but factoring the product back into primes is extremely hard. A 2048-bit RSA key uses primes with about 300 digits each.
Hash tables	Hash table sizes are often chosen as primes to minimize collisions. A prime modulus distributes keys more uniformly across buckets.
Error detection	Cyclic redundancy checks (CRC) use prime polynomials. ISBN check digits use modular arithmetic with prime bases.
Random number generation	Linear congruential generators use prime moduli to maximize the period (cycle length) of pseudo-random sequences.

FAQ - Frequently Asked Questions

Is 1 a prime number?

No. By modern convention, 1 is neither prime nor composite. The definition requires exactly two distinct positive divisors (1 and itself). The number 1 has only one divisor (itself), so it does not qualify. This convention keeps the fundamental theorem of arithmetic clean - every integer has a unique prime factorization.

Is there a largest prime number?

No. Euclid proved around 300 BC that there are infinitely many primes. The proof is elegant: assume finitely many primes p1, p2, ... pn. Compute N = p1 x p2 x ... x pn + 1. N is not divisible by any pi (remainder is always 1), so N either is prime itself or has a prime factor not in the list - contradiction. As of 2024, the largest known prime is a Mersenne prime with over 41 million digits.

What are twin primes?

Twin primes are pairs of primes that differ by 2: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43)... The twin prime conjecture - that there are infinitely many such pairs - remains unproven, making it one of the oldest open problems in mathematics.

How does RSA encryption use primes?

RSA picks two large primes p and q, computes n = p x q (the public modulus), and derives encryption/decryption keys from Euler's totient of n. Security relies on the difficulty of factoring n back into p and q. With current technology, factoring a 2048-bit number would take longer than the age of the universe using classical computers.

What is the Sieve of Eratosthenes?

An ancient algorithm (circa 240 BC) for finding all primes up to a limit N. Start with a list of numbers 2 to N. Take the first unmarked number (2), mark all its multiples as composite. Move to the next unmarked number (3), mark its multiples. Repeat until you reach the square root of N. All remaining unmarked numbers are prime. Time complexity: O(N log log N).

How many primes are there below 1 million?

There are 78,498 primes below 1,000,000. The prime counting function pi(x) grows approximately as x / ln(x) - this is the Prime Number Theorem. So about 1 in every 14 numbers near 1 million is prime, compared to 1 in 4 near 10.

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