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Prime numbers
Testing for primality by trial division up to the square root.
A prime has no divisors except 1 and itself. To test n, try the primes up to √n only: if n had a factor larger than √n, it would also have one smaller. For enormous numbers this is too slow and probabilistic tests take over, but the idea is the same.
مثال کار شده · is 97 prime
قدم به قدم
- \sqrt{97} \approx 9
Only primes up to √97 need testing: any factor above it pairs with one below.
- 97 \bmod 2 = 1
2 does not divide 97.
- 97 \bmod 3 = 1
3 does not divide 97.
- 97 \bmod 5 = 2
5 does not divide 97.
- 97 \bmod 7 = 6
7 does not divide 97.
- \text{97 is prime}
No prime up to the square root divides it, so it has no factors other than 1 and itself.
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چطور: Prime numbers
- Only primes up to √97 need testing: any factor above it pairs with one below.
- 2 does not divide 97.
- 3 does not divide 97.
- 5 does not divide 97.
- 7 does not divide 97.
- No prime up to the square root divides it, so it has no factors other than 1 and itself.
سوالاتي که مردم ميپرسن
Why are primes so important?
Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.
How do I tell whether a big number is prime?
Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller-Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.
بیشتر در Number Theory
Prime factorisationGCD and LCMModular arithmeticDivisorsSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory