maths.free › Number Theory › Prime numbers

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.

Contoh soal · is 97 prime

Number-Theory · Prime-Numbers

Is 97 prime

97

Langkah demi langkah

  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.

Tampilkan jawaban
\text{97 is prime}

Sekarang Anda Pilih soal, atau ketik atau gambar soal Anda sendiri. Setiap langkah, sebuah gambar, jawabannya tersembunyi sampai Anda memintanya.

Simpan pekerjaan Anda sendiri

Sebuah akun gratis menambahkan catatan pada setiap pelajaran, catatan apa yang telah Anda selesaikan, masalah yang Anda selesaikan di satu tempat, dan tutor yang dapat Anda tanyakan tentang halaman ini. matematika itu sendiri terbuka untuk semua orang, ditandatangani atau tidak.

Daftar Masuk

Simbol yang digunakan di sini

Sentuh simbol apapun untuk definisi lengkap, gambar, dan apa setiap huruf di dalamnya berarti.

Bagaimana: 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.

Pertanyaan yang sering diajukan

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.

Lainnya di Number Theory

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