Members of the imaginary quadratic field Q(sqrt(-1)) and

form a ring often denoted Z[i], or sometimes k(i) that are prime.

(primes of the form 4n + 3)

_{3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503}

Odd primes p which divide the class

number of the p-th cyclotomic field.

_{37, 59, 67, 101, 103, 131, 149, 157, 233, 257, 263, 271, 283, 293, 307, 311, 347, 353, 379, 389, 401, 409, 421, 433, 461, 463, 467, 491, 523, 541, 547, 557, 577, 587, 593, 607, 613, 617, 619}

Let d(p) be the shadow of the sequence f(n) = seq^{1-1}(n)

(which gives the number of sequences without repetitions

that can be obtained from n distinct objects),

i.e. the count of sequence entries f(0), f(1), f(2), ...., f(h-1)

divisible by an integer h. If d(p) = 0, then p is this type of prime.

_{3, 7, 11, 17, 47, 53, 61, 67, 73, 79, 89, 101, 139, 151, 157, 191, 199}

Primes p for which the least positive

primitive root is not a primitive root of p^{2}.

_{2, 40487, 6692367337}

Primes p for which the binomial coefficient

(2p - 1)! / [ (p - 1)! ((2p - 1) - (p - 1))! ] ≡ 1 (mod p^{4}).

_{16843, 2124679}

As of 2011, these are the only known Wolstenholme primes.

Of the form p_{n}# −1 or p_{n}# + 1.

_{3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309}

Primes p for which there are no solutions to H_{k} ≡ 0 (mod p) and

H_{k} ≡ −ω_{p} (mod p) for 1 ≤ k ≤ p−2,

where ω_{p} is the Wolstenholme quotient.

_{5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349}

A prime number p which has more solutions to the

equation x − φ(x) = p, than any other integer below p and above one, where, φ is Euler's totient function.

_{2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889}

Of the form p_{n}# + 1 (a subset of primorial primes),

where p_{n}# is the n^{th} primorial, i.e. the product of the first n primes.

_{3, 7, 31, 211, 2311, 200560490131}

Primes that are the number of partitions of a set with n members.

_{2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837}

The next term has 6,539 digits.

Of the form ⌊θ^{3n}⌋, where θ is an eponymous constant.

Its value is unknown, but if the Riemann hypothesis is true it is approximately 1.3063778838630806904686144926...

This form is prime for all positive integers n.

_{2, 11, 1361, 2521008887, 16022236204009818131831320183}

A prime number from an eponymous number sequence that can be used to count certain kinds of binary trees.

The first few numbers are: 0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 98,

207, 451, 983, 2179, 4850, 10905, 24631, 56011, ...

_{2, 3, 11, 23, 983, 2179, 24631, 3626149, 253450711, 596572387}

Primes p such that (p, p−5) is an irregular pair.

_{37}

Primes p such that (p, p−9) is an irregular pair.

_{67, 877}

Those numbers which are prime AND

for which the following process ends in 1:

Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until

the number equals 1 (where it will stay), or

it loops endlessly in a cycle which does not include 1.

_{7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487, 563, 617, 653, 673, 683, 709, 739, 761, 863, 881, 907, 937, 1009, 1033, 1039, 1093}

Primes p_{n} for which p_{n}^{2} > p_{n−i} p_{n+i} for all 1 ≤ i ≤ n−1,

where p_{n} is the n^{th} prime.

_{5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307}

A number that remains prime on any

cyclic rotation of its digits (in base 10).

_{2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331}

Some sources (including OEIS) only list the smallest prime

in each cycle, for example listing 13 but omitting 31.

A member of the sequence G_{n} that satisfies the relation:

2t / ( e^{t} + 1 ) = SUMMATION [from n=1 it infinity] [ G_{n} ( t^{n} / n! )]

that is also prime.

17 is conjectured to be the only such prime.

A prime divisor of the order of the Monster group M,

the largest of the sporadic simple groups.

There are precisely 15 such primes: all 15 are Chen primes.

_{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}

Primes of the form n ʅ ± 1, where n ʅ denotes the swinging factorial, which is defined in terms of the double swinging factorial as

n ʅ = (n-1) ʅ ʅ n ʅ ʅ and

n ʅ ʅ = 1 for n ≤ 0 and

n ʅ ʅ = (n - 2) ʅ ʅ n^{[ n odd ]} ( 4 / n )^{[ n even ]} for n > 0.

_{2, 3, 5, 7, 19, 29, 31, 71, 139, 251, 631, 3433, 12011}

Of the form 2^{u}3^{v} + 1 for some integers u,v ≥ 0.

These are also class 1- primes.

_{2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, 1153, 1297, 1459, 2593, 2917, 3457, 3889, 10369, 12289, 17497, 18433, 39367, 52489, 65537, 139969, 147457}

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).

_{3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991}

A prime number, n, such that when a Fibonacci sequence is formed with the first term equal to the absolute value of the successive

differences between consecutive digits of n and the second term equal to the sum of the decimal digits of n, n itself appears as a term in this Fibonacci sequence.

_{29, 683, 997, 2207, 30571351}

Primes p for which, ( b^{p-1} - 1) / p

in a given base b, gives a cyclic number.

Primes p for base 10:

_{7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 313, 337, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593}

Primes p which do not divide the class

number of the p-th cyclotomic field.

_{3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 53, 61, 71, 73, 79, 83, 89, 97, 107, 109, 113, 127, 137, 139, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 239, 241, 251, 269, 277, 281}

A prime p > 5 if p^{2} divides the Fibonacci number F_{p - (p/5)},

where the Legendre symbol (p/5) is defined as

(p/5) = 1 for p ≡ ± 1 (mod 5) and

(p/5) = - 1 for p ≡ ± 2 (mod 5).

As of 2013, no Wall-Sun-Sun primes are known.

Primes with a prime index in the sequence of prime numbers

(the 2nd, 3rd, 5th, ... prime).

_{3, 5, 11, 17, 31, 41, 59, 67, 83, 109, 127, 157, 179, 191, 211, 241, 277, 283, 331, 353, 367, 401, 431, 461, 509, 547, 563, 587, 599, 617, 709, 739, 773, 797, 859, 877, 919, 967, 991}

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