The first 1000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms.
The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×1018.
There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes.
A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2×1022) smaller than 1024, if the Riemann hypothesis is true.
[4] Below are listed the first prime numbers of many named forms and types.
n is a natural number (including 0) in the definitions.
Primes that are the number of partitions of a set with n members.
A circular prime number is a number that remains prime on any cyclic rotation of its digits (in base 10).
Some sources only list the smallest prime in each cycle, for example, listing 13, but omitting 31 (OEIS really calls this sequence circular primes, but not the above sequence):
Primes that having any one of their (base 10) digits changed to any other value will always result in a composite number.
Eisenstein integers that are irreducible and real numbers (primes of the form 3n − 1).
The name "emirp" is the reverse of the word "prime".
Of the form pn# + 1 (a subset of primorial primes).
The probability of the existence of another Fermat prime is less than one in a billion.
[6] Of the form a2n + 1 for fixed integer a. a = 2: 3, 5, 17, 257, 65537 (OEIS: A019434) a = 4: 5, 17, 257, 65537 a = 6: 7, 37, 1297 a = 8: (none exist) a = 10: 11, 101 a = 12: 13 a = 14: 197 a = 16: 17, 257, 65537 a = 18: 19 a = 20: 401, 160001 a = 22: 23 a = 24: 577, 331777 Primes in the Fibonacci sequence F0 = 0, F1 = 1, Fn = Fn−1 + Fn−2.
Primes p for which there are no solutions to Hk ≡ 0 (mod p) and Hk ≡ −ωp (mod p) for 1 ≤ k ≤ p−2, where Hk denotes the k-th harmonic number and ωp denotes the Wolstenholme quotient.
Primes p for which p − 1 divides the square of the product of all earlier terms.
Odd primes p that divide the class number of the p-th cyclotomic field.
They are also called full reptend primes.
Primes in the Lucas number sequence L0 = 2, L1 = 1, Ln = Ln−1 + Ln−2.
All Mersenne primes are, by definition, members of this sequence.
This include the following: Of the form ⌊θ3n⌋, where θ is Mills' constant.
This form is prime for all positive integers n.
Primes that remain the same when their decimal digits are read backwards.
Primes in the Pell number sequence P0 = 0, P1 = 1, Pn = 2Pn−1 + Pn−2.
Primes p that do not divide the class number of the p-th cyclotomic field.
Primes that cannot be generated by any integer added to the sum of its decimal digits.
As of 2011[update], these are the only known Stern primes, and possibly the only existing.
The list of primes p for which the period length of the decimal expansion of 1/p is unique (no other prime gives the same period).
Primes p such that ap − 1 ≡ 1 (mod p2) for fixed integer a > 1.