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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.

tusaale · is 97 prime

Number-Theory · Prime-Numbers

Is 97 prime

97

tallaabo tallaabo

  1. \sqrt{97} \approx 9

    Only primes up to √97 need testing: any factor above it pairs with one below.

  2. 97 \bmod 2 = 1

    2 does not divide 97.

  3. 97 \bmod 3 = 1

    3 does not divide 97.

  4. 97 \bmod 5 = 2

    5 does not divide 97.

  5. 97 \bmod 7 = 6

    7 does not divide 97.

  6. \text{97 is prime}

    No prime up to the square root divides it, so it has no factors other than 1 and itself.

Jawaabta muuji
\text{97 is prime}

Haddaad Dooro dhibaato, ama qoraal ama sawirkaaga. Talaab kasta, sawir, jawaabta qarsoon ilaa aad weydiiso.

Ku hay shaqadaada

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Ku soo biir Ku soo gal

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Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

Sida loo: Prime numbers

  1. Only primes up to √97 need testing: any factor above it pairs with one below.
  2. 2 does not divide 97.
  3. 3 does not divide 97.
  4. 5 does not divide 97.
  5. 7 does not divide 97.
  6. No prime up to the square root divides it, so it has no factors other than 1 and itself.

Su'aalaha dadku waydiiyaan

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.

In ka badan Number Theory

Prime factorisationGCD and LCMModular arithmeticDivisorsSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory