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An example of the trial division algorithm, using successive integers as trial factors, is as follows (in Python): def trial_division ( n : int ) -> list [ int ]: """Return a list of the prime factors for a natural number.""" a = [] # Prepare an empty list. f = 2 # The first possible factor.
In the example given above that is achieved on identifying 11 as next prime, giving a list of all primes less than or equal to 80. Note that numbers that will be discarded by a step are still used while marking the multiples in that step, e.g., for the multiples of 3 it is 3 × 3 = 9 , 3 × 5 = 15 , 3 × 7 = 21 , 3 × 9 = 27 , ..., 3 × 15 = 45 ...
An example of a decision problem is deciding with the help of an algorithm whether a given natural number is prime. Another example is the problem, "given two numbers x and y, does x evenly divide y?" A method for solving a decision problem, given in the form of an algorithm, is called a decision procedure for that problem.
A primality test is an algorithm for determining whether an input number is prime.Among other fields of mathematics, it is used for cryptography.Unlike integer factorization, primality tests do not generally give prime factors, only stating whether the input number is prime or not.
Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1 ( mod p ) . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds.
A prime sieve works by creating a list of all integers up to a desired limit and progressively removing composite numbers (which it directly generates) until only primes are left. This is the most efficient way to obtain a large range of primes; however, to find individual primes, direct primality tests are more efficient [ citation needed ] .
It is known that the prime number theorem gives an accurate count of the primes within short intervals, either unconditionally [5] or based on the Riemann hypothesis, [6] but the lengths of the intervals for which this has been proven are longer than the intervals between consecutive squares, too long to prove Legendre's conjecture.
The following is pseudocode which combines Atkin's algorithms 3.1, 3.2, and 3.3 [1] by using a combined set s of all the numbers modulo 60 excluding those which are multiples of the prime numbers 2, 3, and 5, as per the algorithms, for a straightforward version of the algorithm that supports optional bit-packing of the wheel; although not specifically mentioned in the referenced paper, this ...