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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.

Робочий приклад: is 97 prime

Number-Theory · Prime-Numbers

Is 97 prime

97

Крок за кроком

  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.

Показати відповідь
\text{97 is prime}

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\approx
approximately equal
Equal to the precision shown, not exactly.
a \bmod n
remainder
What is left after dividing a by n.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
\varphi(n),\ \pi(x)
Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.

How to: 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.

Questions people ask

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.

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