Primality

The Lucas–Lehmer primality test (LLT) is an efficient primality test that greatly aids this task.

Although much more efficient methods of primality testing are known, the existence of any effective method is enough to establish decidability.

An integer factorization algorithm can decide primality.

Faulty key generation seeAlso Finding the large primes p and q is usually done by testing random numbers of the right size with probabilistic primality tests which quickly eliminate virtually all non-primes.

However, deterministic primality tests are slow.

Many mathematicians have worked on primality tests for large numbers, often restricted to specific number forms.

More generally, numbers of the form without the primality requirement are called Mersenne numbers.

One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for particular numbers.

Other operations on natural numbers Exponentiation and primality testing are primitive recursive.

Special-purpose algorithms and the largest known prime further In addition to the aforementioned tests applying to any natural number n, a number of much more efficient primality tests is available for special numbers.

Testing primality and integer factorization There are various methods to determine whether a given number n is prime.

The contrary is not true: for a primality test it is enough to show that a factor exists when checking a composite number.

The ease of primality testing is a crucial part of the RSA algorithm, as it is necessary to find large prime numbers to start with.

Therefore, it is generally used before starting a proof of primality.

The usual algorithm to generate prime numbers is to generate random odd numbers and test them for primality.

This makes tests based on Fermat's Little Theorem risky compared to other more stringent tests such as the Solovay-Strassen primality test or a strong pseudoprime test.

We can use a primality testing algorithm to make sure that is indeed composite.