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.
अद्यतनित तपासलेली उदाहरणे: ६
360 = 2³ × 3² × 5, so it is composite with 24 divisors adding up to 1170. The nearest primes are 359 and 367.
| Divisor | Paired divisor n ÷ d |
|---|---|
| 1 | 360 |
| 2 | 180 |
| 3 | 120 |
| 4 | 90 |
| 5 | 72 |
| 6 | 60 |
| 8 | 45 |
| 9 | 40 |
| 10 | 36 |
| 12 | 30 |
| 15 | 24 |
| 18 | 20 |
Each prime is divided out as many times as it goes before moving on.
Candidates are tested with the same Miller–Rabin check.
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.
पडताळणीचा स्रोत: Python 3.8 trial division and divisor enumeration (scratchpad forkA/verify.py)
पडताळणीचा स्रोत: Project Euler #3 (largest prime factor 6857); product and primality checked in Python 3.8
पडताळणीचा स्रोत: F. N. Cole, 1903 (Bull. AMS 10:134); product and primality of both factors checked by Python trial division
पडताळणीचा स्रोत: Table of primes (OEIS A000040): 89, 97, 101
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.
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.
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.
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.
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.
अचूकता तुमच्या इनपुटवर आणि पद्धतीच्या गृहीतकांवर अवलंबून असते. दशांश गणना 50 सार्थ अंक वापरते, पण अंदाज, संख्यात्मक पद्धती आणि मूळ डेटा कमी अचूक असू शकतात; दाखवलेल्या मूल्यांचे पूर्णांकन केल्याने या मर्यादा दूर होत नाहीत. स्वतंत्र स्रोतांतील सोडवलेल्या उदाहरणांशी पडताळणी: ६. उदाहरणार्थ, “360” ची पडताळणी Python 3.8 trial division and divisor enumeration (scratchpad forkA/verify.py) याच्याशी केली आहे.
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.
या कॅल्क्युलेटरमध्ये स्वतंत्र स्रोतांमधील उत्तरांसह ६ सोडवलेली उदाहरणे आहेत. ती चाचणी संचात चालतात आणि तुम्ही ती येथेही चालवू शकता.
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