List of prime numbers
Template:Short description Template:Use dmy dates {{#invoke:Hatnote|hatnote}}{{#ifeq:||}} This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.
The first 1,000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. The number 1 is neither prime nor composite.
The first 1,000 prime numbers
The following table lists the first 1,000 primes, with 20 columns of consecutive primes in each of the 50 rows.<ref>Template:Cite book</ref>
(sequence A000040 in the OEIS).
The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×1018.<ref>Tomás Oliveira e Silva, Goldbach conjecture verification Template:Webarchive. Retrieved 16 July 2013</ref> That means 95,676,260,903,887,607 primes<ref>(sequence A080127 in the OEIS)</ref> (nearly 1017), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2Template:E) smaller than 1023. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2Template:E) smaller than 1024, if the Riemann hypothesis is true.<ref name="Franke">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Lists of primes by type
Below are listed the first prime numbers of many named forms and types. More details are in the article for the name. n is a natural number (including 0) in the definitions.
Balanced primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Balanced primes are primes with equal-sized prime gaps before and after them, making them the arithmetic mean of their next larger and next smaller prime.
- 5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393 (Template:OEIS2C).
Bell primes
Template:See also Bell primes are primes that are also the number of partitions of some finite set.
2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. The next term has 6,539 digits. (Template:OEIS2C)
Chen primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Chen primes are primes p such that p+2 is either a prime or semiprime.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, 107, 109, 113, 127, 131, 137, 139, 149, 157, 167, 179, 181, 191, 197, 199, 211, 227, 233, 239, 251, 257, 263, 269, 281, 293, 307, 311, 317, 337, 347, 353, 359, 379, 389, 401, 409 (Template:OEIS2C)
Circular primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A circular prime is a number that remains prime on any cyclic rotation of its base 10 digits.
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 (Template:OEIS2C)
Some sources only include the smallest prime in each cycle. For example, listing 13, but omitting 31.
2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111 (Template:OEIS2C)
Cluster primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p.
3, 5, 7, 11, 13, 17, 19, 23, ... (Template:OEIS2C)
All primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are:
2, 97, 127, 149, 191, 211, 223, 227, 229, 251.
Cousin primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:See also
Cousin primes are pairs of primes that differ by four.
(3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), (79, 83), (97, 101), (103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281) (Template:OEIS2C, Template:OEIS2C)
Cuban primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Cuban primes are primes <math>p</math> of the form <math>p = k^3 - (k - 1)^3,</math> where <math>k</math> is a natural number.
7, 19, 37, 61, 127, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 2791, 3169, 3571, 4219, 4447, 5167, 5419, 6211, 7057, 7351, 8269, 9241, 10267, 11719, 12097, 13267, 13669, 16651, 19441, 19927, 22447, 23497, 24571, 25117, 26227, 27361, 33391, 35317 (Template:OEIS2C)
The term is also used to refer to primes <math>p</math> of the form <math>p = (k^3 - (k - 2)^3)/2,</math> where <math>k</math> is a natural number.
13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249 (Template:OEIS2C)
Cullen primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Cullen primes are primes p of the form p=k2k + 1, for some natural number k.
3, 393050634124102232869567034555427371542904833 (Template:OEIS2C)
Delicate primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Delicate primes are those primes that always become a composite number when any of their base 10 digit is changed.
294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139 (Template:OEIS2C)
Dihedral primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Dihedral primes are primes that satisfy 180° rotational symmetry and mirror symmetry on a seven-segment display.
2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 (Template:OEIS2C)
Real Eisenstein primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Real Eisenstein primes are real Eisenstein integers that are irreducible. Equivalently, they are primes of the form 3k − 1, for a positive integer k.
2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137, 149, 167, 173, 179, 191, 197, 227, 233, 239, 251, 257, 263, 269, 281, 293, 311, 317, 347, 353, 359, 383, 389, 401 (Template:OEIS2C)
Emirps
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Emirps are those primes that become a different prime after their base 10 digits have been reversed. The name "emirp" is the reverse of the word "prime".
13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991 (Template:OEIS2C)
Euclid primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Euclid primes are primes p such that p−1 is a primorial.
3, 7, 31, 211, 2311, 200560490131 (Template:OEIS2C<ref name="A018239">Template:OEIS2C includes 2 = empty product of first 0 primes plus 1, but 2 is excluded in this list.</ref>)
Euler irregular primes
Template:See Euler irregular primes are primes <math>p</math> that divide an Euler number <math>E_{2n},</math> for some <math>0\leq 2n\leq p-3.</math>
19, 31, 43, 47, 61, 67, 71, 79, 101, 137, 139, 149, 193, 223, 241, 251, 263, 277, 307, 311, 349, 353, 359, 373, 379, 419, 433, 461, 463, 491, 509, 541, 563, 571, 577, 587 (Template:OEIS2C)
Euler (p, p − 3) irregular primes
Euler (p, p - 3) irregular primes are primes p that divide the (p + 3)rd Euler number.
149, 241, 2946901 (Template:OEIS2C)
Factorial primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Factorial primes are primes whose distance to the next factorial number is one.
2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 (Template:OEIS2C)
Fermat primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Fermat primes are primes p of the form p = 22k + 1, for a non-negative integer k. Template:As of only five Fermat primes have been discovered.
3, 5, 17, 257, 65537 (Template:OEIS2C)
Generalized Fermat primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Generalized Fermat primes are primes p of the form p = a2k + 1, for a non-negative integer k and even natural number a.
| <math>a</math> | Generalized Fermat primes with base a |
|---|---|
| 2 | 3, 5, 17, 257, 65537, ... (Template:OEIS2C) |
| 4 | 5, 17, 257, 65537, ... |
| 6 | 7, 37, 1297, ... |
| 8 | (none exist) |
| 10 | 11, 101, ... |
| 12 | 13, ... |
| 14 | 197, ... |
| 16 | 17, 257, 65537, ... |
| 18 | 19, ... |
| 20 | 401, 160001, ... |
| 22 | 23, ... |
| 24 | 577, 331777, ... |
Fibonacci primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Fibonacci primes are primes that appear in the Fibonacci sequence.
2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 (Template:OEIS2C)
Fortunate primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Fortunate primes are primes that are also Fortunate numbers. There are no known composite Fortunate numbers.Template:Citation needed
3, 5, 7, 13, 17, 19, 23, 37, 47, 59, 61, 67, 71, 79, 89, 101, 103, 107, 109, 127, 151, 157, 163, 167, 191, 197, 199, 223, 229, 233, 239, 271, 277, 283, 293, 307, 311, 313, 331, 353, 373, 379, 383, 397 (Template:OEIS2C)
Gaussian primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Gaussian primes are primes p of the form p = 4k + 3, for a non-negative integer k.
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 (Template:OEIS2C)
Good primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Good primes are primes p satisfying ab < p2, for all primes a and b such that a,b < p
5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307 (Template:OEIS2C)
Happy primes
Template:See also Happy primes are primes that are also happy numbers.
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 (Template:OEIS2C)
Harmonic primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Harmonic primes are 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.<ref>Template:Cite journal</ref>
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 (Template:OEIS2C)
Higgs primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Higgs primes are primes p for which p − 1 divides the square of the product of all smaller Higgs primes.
2, 3, 5, 7, 11, 13, 19, 23, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 79, 101, 107, 127, 131, 139, 149, 151, 157, 173, 181, 191, 197, 199, 211, 223, 229, 263, 269, 277, 283, 311, 317, 331, 347, 349 (Template:OEIS2C)
Highly cototient primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Highly cototient primes are primes that are a cototient more often than any integer below it except 1.
2, 23, 47, 59, 83, 89, 113, 167, 269, 389, 419, 509, 659, 839, 1049, 1259, 1889 (Template:OEIS2C)
Home primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} For Template:Math, write the prime factorization of Template:Mvar in base 10 and concatenate the factors; iterate until a prime is reached.
For a non-negative integer, its home prime is obtained by concatenating its prime factors in increasing order repeatedly, until a prime is achieved.
2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277 (Template:OEIS2C)
Irregular primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Irregular primes are odd primes p that 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 (Template:OEIS2C)
(p, p − 3) irregular primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The (p, p - 3) irregular primes are primes p such that (p, p − 3) is an irregular pair.
16843, 2124679 (Template:OEIS2C)
(p, p − 5) irregular primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The (p, p - 5) irregular primes are primes p such that (p, p − 5) is an irregular pair.<ref name="Johnson">Template:Cite journal</ref>
(p, p − 9) irregular primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The (p, p - 9) irregular primes are primes p such that (p, p − 9) is an irregular pair.<ref name="Johnson" />
67, 877 (Template:OEIS2C)
Isolated primes
Template:See Isolated primes are primes p such that both p − 2 and p + 2 are both composite.
2, 23, 37, 47, 53, 67, 79, 83, 89, 97, 113, 127, 131, 157, 163, 167, 173, 211, 223, 233, 251, 257, 263, 277, 293, 307, 317, 331, 337, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 439, 443, 449, 457, 467, 479, 487, 491, 499, 503, 509, 541, 547, 557, 563, 577, 587, 593, 607, 613, 631, 647, 653, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 839, 853, 863, 877, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 (Template:OEIS2C)
Leyland primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Leyland primes are primes p of the form p = ab + ba, where a and b are integers larger than one.
17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (Template:OEIS2C)
Long primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Long primes, or full reptend primes, are odd primes p for which <math>(10^{p-1}-1)/p</math> is a cyclic number. Bases other than 10 are also used.
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 (Template:OEIS2C)
Lucas primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Lucas primes are primes that appear in the Lucas sequence.
2,<ref>It varies whether L0 = 2 is included in the Lucas numbers.</ref> 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 (Template:OEIS2C)
Lucky primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Lucky primes are primes that are also lucky numbers.
3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997 (Template:OEIS2C)
Mersenne primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Mersenne primes are primes p of the form p = 2k − 1, for some non-negative integer k.
3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (Template:OEIS2C)
Template:As of, there are 52 known Mersenne primes.Template:Citation needed The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits.Template:Citation needed The largest known prime 2136,279,841−1 is the 52nd Mersenne prime.Template:Citation needed
Mersenne divisors
Mersenne divisors are primes that divide 2k − 1, for some prime k. Every Mersenne prime p is also a Mersenne divisor, with k = p.
3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343 (Template:OEIS2C)
Mersenne prime exponents
Primes p such that 2p − 1 is prime.
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917 (Template:OEIS2C)
Template:As of, two more are known to be in the sequence, but it is not known whether they are the next:
82589933, 136279841
Double Mersenne primes
Template:See A subset of Mersenne primes of the form 22p−1 − 1 for prime p.
7, 127, 2147483647, 170141183460469231731687303715884105727 (primes in Template:OEIS2C)
Generalized repunit primes
Of the form (an − 1) / (a − 1) for fixed integer a.
For a = 2, these are the Mersenne primes, while for a = 10 they are the repunit primes. For other small a, they are given below:
a = 3: 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (Template:OEIS2C)
a = 4: 5 (the only prime for a = 4)
a = 5: 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 (Template:OEIS2C)
a = 6: 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 (Template:OEIS2C)
a = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457
a = 8: 73 (the only prime for a = 8)
a = 9: none exist
Other generalizations and variations
Many generalizations of Mersenne primes have been defined. This include the following:
- Primes of the form Template:Math,<ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref><ref>Template:Cite OEIS</ref> including the Mersenne primes and the cuban primes as special cases
- Williams primes, of the form Template:Math
Mills primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form ⌊θ3n⌋, where θ is Mills' constant. This form is prime for all positive integers n.
2, 11, 1361, 2521008887, 16022236204009818131831320183 (Template:OEIS2C)
Minimal primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes:
2, 3, 5, 7, 11, 19, 41, 61, 89, 409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 (Template:OEIS2C)
Newman–Shanks–Williams primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Newman–Shanks–Williams numbers that are prime.
7, 41, 239, 9369319, 63018038201, 489133282872437279, 19175002942688032928599 (Template:OEIS2C)
Non-generous primes
Primes p for which the least positive primitive root is not a primitive root of p2. Three such primes are known; it is not known whether there are more.<ref>Template:Cite journal</ref>
2, 40487, 6692367337 (Template:OEIS2C)
Palindromic primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes that remain the same when their decimal digits are read backwards.
2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 (Template:OEIS2C)
Palindromic wing primes
Primes of the form <math>\frac{a \big( 10^m-1 \big)}{9} \pm b \times 10^{\frac{ m-1 }{2}}</math> with <math>0 \le a \pm b < 10</math>.<ref>Template:Cite journal</ref> This means all digits except the middle digit are equal.
101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 (Template:OEIS2C)
Partition primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Partition function values that are prime.
2, 3, 5, 7, 11, 101, 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 (Template:OEIS2C)
Pell primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes in the Pell number sequence P0 = 0, P1 = 1, Pn = 2Pn−1 + Pn−2.
2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 (Template:OEIS2C)
Permutable primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Any permutation of the decimal digits is a prime.
2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111 (Template:OEIS2C)
Perrin primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes in the Perrin number sequence P(0) = 3, P(1) = 0, P(2) = 2, P(n) = P(n−2) + P(n−3).
2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 (Template:OEIS2C)
Pierpont primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form 2u3v + 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 (Template:OEIS2C)
Pillai primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes p for which there exist n > 0 such that p divides n! + 1 and n does not divide p − 1.
23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, 227, 233, 239, 251, 257, 269, 271, 277, 293, 307, 311, 317, 359, 379, 383, 389, 397, 401, 419, 431, 449, 461, 463, 467, 479, 499 (Template:OEIS2C)
Primes of the form n4 + 1
Of the form n4 + 1.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001 (Template:OEIS2C)
Primeval primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number.
2, 13, 37, 107, 113, 137, 1013, 1237, 1367, 10079 (Template:OEIS2C)
Primorial primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form pn# ± 1.
3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of Template:OEIS2C and Template:OEIS2C<ref name="A018239"/>)
Proth primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form k×2n + 1, with odd k and k < 2n.
3, 5, 13, 17, 41, 97, 113, 193, 241, 257, 353, 449, 577, 641, 673, 769, 929, 1153, 1217, 1409, 1601, 2113, 2689, 2753, 3137, 3329, 3457, 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (Template:OEIS2C)
Pythagorean primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form 4n + 1.
5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449 (Template:OEIS2C)
Prime quadruplets
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:See also
Where (p, p+2, p+6, p+8) are all prime.
(5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109), (191, 193, 197, 199), (821, 823, 827, 829), (1481, 1483, 1487, 1489), (1871, 1873, 1877, 1879), (2081, 2083, 2087, 2089), (3251, 3253, 3257, 3259), (3461, 3463, 3467, 3469), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) (Template:OEIS2C, Template:OEIS2C, Template:OEIS2C, Template:OEIS2C)
Quartan primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form x4 + y4, where x,y > 0.
2, 17, 97, 257, 337, 641, 881 (Template:OEIS2C)
Ramanujan primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Integers Rn that are the smallest to give at least n primes from x/2 to x for all x ≥ Rn (all such integers are primes).
2, 11, 17, 29, 41, 47, 59, 67, 71, 97, 101, 107, 127, 149, 151, 167, 179, 181, 227, 229, 233, 239, 241, 263, 269, 281, 307, 311, 347, 349, 367, 373, 401, 409, 419, 431, 433, 439, 461, 487, 491 (Template:OEIS2C)
Regular primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes p that 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 (Template:OEIS2C)
Repunit primes
Template:See also Primes containing only the decimal digit 1.
11, 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) (Template:OEIS2C)
The next have 317, 1031, 49081, 86453, 109297, and 270343 digits, respectively (Template:OEIS2C).
Residue classes of primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a.
The primes of the form 2n+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4n+1 are Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes (with 2 omitted). The classes 10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d.
If a and d are relatively prime, the arithmetic progression contains infinitely many primes.
2n+1: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 (Template:OEIS2C)
4n+1: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137 (Template:OEIS2C)
4n+3: 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107 (Template:OEIS2C)
6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139 (Template:OEIS2C)
6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 (Template:OEIS2C)
8n+1: 17, 41, 73, 89, 97, 113, 137, 193, 233, 241, 257, 281, 313, 337, 353 (Template:OEIS2C)
8n+3: 3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251 (Template:OEIS2C)
8n+5: 5, 13, 29, 37, 53, 61, 101, 109, 149, 157, 173, 181, 197, 229, 269 (Template:OEIS2C)
8n+7: 7, 23, 31, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263 (Template:OEIS2C)
10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281 (Template:OEIS2C)
10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 (Template:OEIS2C)
10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277 (Template:OEIS2C)
10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359 (Template:OEIS2C)
12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 (Template:OEIS2C)
12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 (Template:OEIS2C)
12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 (Template:OEIS2C)
12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 (Template:OEIS2C)
Safe primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Where p and (p−1) / 2 are both prime.
5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907 (Template:OEIS2C)
Self primes in base 10
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes that cannot be generated by any integer added to the sum of its decimal digits.
3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873 (Template:OEIS2C)
Sexy primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Where (p, p + 6) are both prime.
(5, 11), (7, 13), (11, 17), (13, 19), (17, 23), (23, 29), (31, 37), (37, 43), (41, 47), (47, 53), (53, 59), (61, 67), (67, 73), (73, 79), (83, 89), (97, 103), (101, 107), (103, 109), (107, 113), (131, 137), (151, 157), (157, 163), (167, 173), (173, 179), (191, 197), (193, 199) (Template:OEIS2C, Template:OEIS2C)
Smarandache–Wellin primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes that are the concatenation of the first n primes written in decimal.
2, 23, 2357 (Template:OEIS2C)
The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.
Solinas primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form 2k − c1·2k−1 − c2·2k−2 − ... − ck.
- 3, 5, 7, 11, 13 (Template:OEIS2C)
- 232 − 5, the largest prime that fits into 32 bits of memory.<ref>{{#invoke:citation/CS1|citation
|CitationClass=web }}</ref>
- 264 − 59, the largest prime that fits into 64 bits of memory.
Sophie Germain primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Where p and 2p + 1 are both prime. A Sophie Germain prime has a corresponding safe prime.
2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953 (Template:OEIS2C)
Stern primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes that are not the sum of a smaller prime and twice the square of a nonzero integer.
2, 3, 17, 137, 227, 977, 1187, 1493 (Template:OEIS2C)
Template:As of, these are the only known Stern primes, and possibly the only existing.
Super-primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes with prime-numbered indexes 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 (Template:OEIS2C)
Supersingular primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} There are exactly fifteen supersingular primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 (Template:OEIS2C)
Thabit primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form 3×2n − 1.
2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 (Template:OEIS2C)
The primes of the form 3×2n + 1 are related.
7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657 (Template:OEIS2C)
Prime triplets
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:See also
Where (p, p+2, p+6) or (p, p+4, p+6) are all prime.
(5, 7, 11), (7, 11, 13), (11, 13, 17), (13, 17, 19), (17, 19, 23), (37, 41, 43), (41, 43, 47), (67, 71, 73), (97, 101, 103), (101, 103, 107), (103, 107, 109), (107, 109, 113), (191, 193, 197), (193, 197, 199), (223, 227, 229), (227, 229, 233), (277, 281, 283), (307, 311, 313), (311, 313, 317), (347, 349, 353) (Template:OEIS2C, Template:OEIS2C, Template:OEIS2C)
Truncatable prime
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
Left-truncatable
Primes that remain prime when the leading decimal digit is successively removed.
2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683 (Template:OEIS2C)
Right-truncatable
Primes that remain prime when the least significant decimal digit is successively removed.
2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797 (Template:OEIS2C)
Two-sided
Primes that are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes:
2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (Template:OEIS2C)
Twin primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Template:See also
Where (p, p+2) are both prime.
(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), (149, 151), (179, 181), (191, 193), (197, 199), (227, 229), (239, 241), (269, 271), (281, 283), (311, 313), (347, 349), (419, 421), (431, 433), (461, 463) (Template:OEIS2C, Template:OEIS2C)
Unique primes
Template:See 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 (Template:OEIS2C)
Wagstaff primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Of the form (2n + 1) / 3.
3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 (Template:OEIS2C)
Values of n:
3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 (Template:OEIS2C)
Wall–Sun–Sun primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A prime p > 5, if p2 divides the Fibonacci number <math>F_{p - \left(\fracTemplate:PTemplate:5\right)}</math>, where the Legendre symbol <math>\left(\fracTemplate:PTemplate:5\right)</math> is defined as
- <math>\left(\frac{p}{5}\right) = \begin{cases} 1 &\textrm{if}\;p \equiv \pm1 \pmod 5\\ -1 &\textrm{if}\;p \equiv \pm2 \pmod 5. \end{cases}</math>
Template:As of, no Wall-Sun-Sun primes have been found below <math>2^{64}</math> (about <math>18\cdot 10^{18}</math>).<ref>Subproject status at PrimeGrid</ref>
Wieferich primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Ambox }}
Primes p such that Template:Nowrap for fixed integer a > 1.
2p − 1 ≡ 1 (mod p2): 1093, 3511 (Template:OEIS2C)
3p − 1 ≡ 1 (mod p2): 11, 1006003 (Template:OEIS2C)<ref>Template:Cite book</ref><ref>{{#invoke:citation/CS1|citation
|CitationClass=web
}}</ref><ref>Template:Cite journal</ref>
4p − 1 ≡ 1 (mod p2): 1093, 3511
5p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 (Template:OEIS2C)
6p − 1 ≡ 1 (mod p2): 66161, 534851, 3152573 (Template:OEIS2C)
7p − 1 ≡ 1 (mod p2): 5, 491531 (Template:OEIS2C)
8p − 1 ≡ 1 (mod p2): 3, 1093, 3511
9p − 1 ≡ 1 (mod p2): 2, 11, 1006003
10p − 1 ≡ 1 (mod p2): 3, 487, 56598313 (Template:OEIS2C)
11p − 1 ≡ 1 (mod p2): 71<ref name="RibenboimWelt">Template:Cite book</ref>
12p − 1 ≡ 1 (mod p2): 2693, 123653 (Template:OEIS2C)
13p − 1 ≡ 1 (mod p2): 2, 863, 1747591 (Template:OEIS2C)<ref name="RibenboimWelt" />
14p − 1 ≡ 1 (mod p2): 29, 353, 7596952219 (Template:OEIS2C)
15p − 1 ≡ 1 (mod p2): 29131, 119327070011 (Template:OEIS2C)
16p − 1 ≡ 1 (mod p2): 1093, 3511
17p − 1 ≡ 1 (mod p2): 2, 3, 46021, 48947 (Template:OEIS2C)<ref name="RibenboimWelt" />
18p − 1 ≡ 1 (mod p2): 5, 7, 37, 331, 33923, 1284043 (Template:OEIS2C)
19p − 1 ≡ 1 (mod p2): 3, 7, 13, 43, 137, 63061489 (Template:OEIS2C)<ref name="RibenboimWelt" />
20p − 1 ≡ 1 (mod p2): 281, 46457, 9377747, 122959073 (Template:OEIS2C)
21p − 1 ≡ 1 (mod p2): 2
22p − 1 ≡ 1 (mod p2): 13, 673, 1595813, 492366587, 9809862296159 (Template:OEIS2C)
23p − 1 ≡ 1 (mod p2): 13, 2481757, 13703077, 15546404183, 2549536629329 (Template:OEIS2C)
24p − 1 ≡ 1 (mod p2): 5, 25633
25p − 1 ≡ 1 (mod p2): 2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
Template:As of, these are all known Wieferich primes with a ≤ 25.
Wilson primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes p for which p2 divides (p−1)! + 1.
5, 13, 563 (Template:OEIS2C)
Template:As of, these are the only known Wilson primes.
Wolstenholme primes
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Primes p for which the binomial coefficient <math>{{2p-1}\choose{p-1}} \equiv 1 \pmod{p^4}.</math>
16843, 2124679 (Template:OEIS2C)
Template:As of, these are the only known Wolstenholme primes.
Woodall primes
Template:See also Of the form n×2n − 1.
7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 (Template:OEIS2C)
See also
Template:Portal Template:Div col
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References
External links
- Lists of Primes at the Prime Pages.
- The Nth Prime Page Nth prime through n=10^12, pi(x) through x=3*10^13, Random primes in same range.
- Interface to a list of the first 98 million primes (primes less than 2,000,000,000)
- {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web
|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:topics/PrimeNumberSequences%7Ctopics/PrimeNumberSequences.html}} |title = Prime Number Sequences |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = none }}
- Selected prime related sequences in OEIS.
- Fischer, R. Thema: Fermatquotient B^(P−1) == 1 (mod P^2) Template:In lang (Lists Wieferich primes in all bases up to 1052)
- {{#invoke:citation/CS1|citation
|CitationClass=web }}Template:Cbignore