is the th prime, beginning with . is the number of primes less than or equal to .
where the series terminates when the product of the first primes exceeds .
There exists a positive constant such that
The Riemann hypothesis (§25.10(i)) is equivalent to the statement that for every ,
If is relatively prime to the modulus , then there are infinitely many primes congruent to .
The number of such primes not exceeding is
where depends only on , and is the Euler totient function (§27.2).
A Mersenne prime is a prime of the form . The largest known prime (2009) is the Mersenne prime . For current records see The Great Internet Mersenne Prime Search.
A pseudoprime test is a test that correctly identifies most composite numbers. For example, if , then is composite. Descriptions and comparisons of pseudoprime tests are given in Bressoud and Wagon (2000, §§2.4, 4.2, and 8.2) and Crandall and Pomerance (2005, §§3.4–3.6).
A Carmichael number is a composite number for which for all . There are infinitely many Carmichael numbers.