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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.
ຕົວຢ່າງທີ່ໄດ້ເຮັດ · is 97 prime
ຂັ້ນຕອນຕໍ່ຂັ້ນຕອນ
- \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.
ເປີດເຜີຍຄຳຕອບ
ປັດຈຸບັນທ່ານ ເລືອກບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມເອງ. ທຸກບາດກ້າວ, ຮູບ, ຄໍາຕອບຖືກລົບຈົນກວ່າເຈົ້າຈະຖາມມັນ.
ຕົວສະແດງທີ່ໃຊ້ຢູ່ທີ່ນີ້
ກົດຕົວອັກສອນໃດໜຶ່ງເພື່ອເບິ່ງຄວາມໝາຍເຕັມ, ຮູບ ແລະ ຕົວອັກສອນແຕ່ລະຕົວໃນມັນໝາຍຄວາມວ່າແນວໃດ.
ວິທີການ: 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.
ຄໍາຖາມທີ່ຄົນຖາມ
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.
ເພີ່ມເຕີມໃນ Number Theory
Prime factorisationGCD and LCMModular arithmeticDivisorsSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory