# Prime factorization calculator

> Prime factorization of any whole number up to 60 digits, with a factor tree, a prime or composite verdict, its divisors and the nearest primes.

Interactive version: https://www.calcopenly.com/math/prime-factorization-calculator
Subject: Math calculators

By the fundamental theorem of arithmetic, every whole number above 1 is a product of primes in exactly one way, apart from order: 360 = 2³ × 3² × 5. The calculator divides out primes below 10,000 by trial division, splits larger composite factors with Brent's version of Pollard's rho method, and proves each factor prime with a Miller–Rabin test on the first 13 prime bases, which is deterministic below 3.3 × 10²⁴ (Sorenson and Webster, 2015).

Simplifying fractions and radicals, finding common denominators, and number puzzles all start from a factorization. The exponents also count the divisors: 360 has (3 + 1)(2 + 1)(1 + 1) = 24 divisors that sum to 1,170, and the nearest primes are 359 and 367.

Numbers up to 60 digits are accepted, including expressions such as 2^67 − 1. Above 3.3 × 10²⁴ a prime verdict is labeled “probable prime”, with an error chance below 4⁻²⁰.

## Inputs

- **Whole number**: Up to 60 digits. Expressions work too, e.g. 2^67 − 1.

## Results

- Prime factorization — main result
- Prime or composite
- Number of divisors
- Sum of divisors σ(n)
- Previous prime
- Next prime

## Formula

$$
\begin{gathered} n = p_1^{e_1} p_2^{e_2}\cdots p_k^{e_k} \\[6pt] d(n) = \prod (e_i + 1) \\[6pt] \sigma(n) = \prod \frac{p_i^{e_i+1} - 1}{p_i - 1} \end{gathered}
$$

## Worked examples

### 360

- Whole number: 360
- **Prime factorization: 2³ × 3² × 5**
- **Prime or composite: Composite**
- **Number of divisors: 24**
- **Sum of divisors σ(n): 1,170**
- **Previous prime: 359**
- **Next prime: 367**
- Checked against: Python 3.8 trial division and divisor enumeration (scratchpad forkA/verify.py)

### Project Euler problem 3

- Whole number: 600851475143
- **Prime factorization: 71 × 839 × 1471 × 6857**
- **Number of divisors: 16**
- Checked against: Project Euler #3 (largest prime factor 6857); product and primality checked in Python 3.8

### Cole's factorization of 2^67 − 1

- Whole number: 2^67-1
- **Prime factorization: 193707721 × 761838257287**
- **Prime or composite: Composite**
- **Number of divisors: 4**
- Checked against: F. N. Cole, 1903 (Bull. AMS 10:134); product and primality of both factors checked by Python trial division

### 97 is prime

- Whole number: 97
- **Prime factorization: 97**
- **Prime or composite: Prime**
- **Number of divisors: 2**
- **Sum of divisors σ(n): 98**
- **Previous prime: 89**
- **Next prime: 101**
- Checked against: Table of primes (OEIS A000040): 89, 97, 101

### n = 1 (edge)

- Whole number: 1
- **Prime factorization: 1**
- **Prime or composite: Neither prime nor composite**
- **Number of divisors: 1**
- **Sum of divisors σ(n): 1**
- **Next prime: 2**
- Checked against: Definition: 1 has no prime factors and one divisor

### Mersenne prime 2^61 − 1

- Whole number: 2^61-1
- **Prime factorization: 2305843009213693951**
- **Prime or composite: Prime**
- **Number of divisors: 2**
- Checked against: M61 proved prime by Pervushin (1883), OEIS A000668

## Questions

### How do you find the prime factorization of a number?

Divide by the smallest prime that goes in, and repeat on the quotient until it reaches 1. For 360: 360 ÷ 2 = 180, ÷ 2 = 90, ÷ 2 = 45, ÷ 3 = 15, ÷ 3 = 5, and 5 is prime, so 360 = 2³ × 3² × 5. Only primes up to the square root of what is left need testing; if none divides it, the remainder is itself prime.

### Is 1 a prime number?

No. A prime has exactly two divisors, 1 and itself, and 1 has only one. Excluding 1 keeps factorizations unique: if 1 counted as prime, 6 could be written as 2 × 3, 1 × 2 × 3, 1 × 1 × 2 × 3 and so on. 1 is neither prime nor composite, and 2 is the smallest prime and the only even one.

### How do you find the number of divisors from the prime factorization?

Add 1 to each exponent and multiply. 360 = 2³ × 3² × 5¹, so it has (3 + 1)(2 + 1)(1 + 1) = 24 divisors, because each divisor uses 0 to 3 twos, 0 to 2 threes and 0 or 1 five. The sum of the divisors is a similar product: σ(360) = (2⁴ − 1)/1 × (3³ − 1)/2 × (5² − 1)/4 = 15 × 13 × 6 = 1,170.

### How can you tell if a large number is prime?

Trial division up to √n is far too slow for 20-digit numbers, so the Miller–Rabin test checks a handful of bases instead. Below 3.3 × 10²⁴, passing with the first 13 primes (2 to 41) as bases proves primality (Sorenson and Webster, 2015). 2⁶¹ − 1 = 2,305,843,009,213,693,951 passes and is prime; 2⁶⁷ − 1 fails and equals 193,707,721 × 761,838,257,287.

### Why does prime factorization matter in cryptography?

RSA encryption relies on multiplying two large primes being quick while factoring their product is impractically slow. NIST SP 800-57 rates a 2048-bit RSA modulus, about 617 decimal digits, as giving 112 bits of security. That is ten times the 60-digit limit here, and even within that limit Pollard's rho splits a number quickly only when one of its factors is fairly small.

### How accurate is the prime factorization calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “360” is checked against Python 3.8 trial division and divisor enumeration (scratchpad forkA/verify.py).

### Where does the method come from?

Hardy & Wright, An Introduction to the Theory of Numbers, §2.10 and §16.7 (divisor functions); Sorenson & Webster (2015), Strong pseudoprimes to twelve prime bases; Brent (1980), An improved Monte Carlo factorization algorithm.

## Sources

- Hardy & Wright, An Introduction to the Theory of Numbers, §2.10 and §16.7 (divisor functions)
- [Sorenson & Webster (2015), Strong pseudoprimes to twelve prime bases](https://arxiv.org/abs/1509.00864)
- [Brent (1980), An improved Monte Carlo factorization algorithm](https://maths-people.anu.edu.au/~brent/pub/pub051.html)