List of prime numbers facts for kids

Kids Encyclopedia Facts

A prime number is a special kind of natural number (a counting number like 1, 2, 3, and so on) that is bigger than 1. What makes it special? It can only be divided evenly by two numbers: 1 and itself. For example, 7 is a prime number because you can only divide it by 1 and 7. But 6 is not prime because you can divide it by 1, 2, 3, and 6.

Mathematicians have known for a very long time that there are an endless number of prime numbers. This was proven by a famous Greek mathematician named Euclid. The number 1 is unique; it's not considered a prime number and it's not a composite number (a number that has more than two divisors).

What are Prime Numbers?
Types of Prime Numbers
See also

What are Prime Numbers?

Imagine you have a bunch of items, and you want to arrange them into equal rows and columns.

If you have 7 items, you can only arrange them in one row of 7, or 7 rows of 1. That's why 7 is a prime number!
If you have 6 items, you can arrange them in 1 row of 6, 6 rows of 1, 2 rows of 3, or 3 rows of 2. Because it has more ways to be arranged, 6 is not a prime number.

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

Types of Prime Numbers

Mathematicians love to find patterns and special types of prime numbers. Here are some interesting ones:

Balanced Primes

A balanced prime is a prime number that sits exactly in the middle of the prime numbers before and after it. It's like being the middle kid in a family of primes!

For example, 5 is a balanced prime because the prime before it is 3, and the prime after it is 7. (3 + 7) / 2 = 5.
First few: 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.

Bell Primes

Bell primes are prime numbers that represent the number of ways you can divide a set of items into smaller, non-empty groups.

First few: 2, 5, 877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837.

Chen Primes

A Chen prime is a prime number p where p + 2 is either another prime number or a semiprime (a number that is the product of two primes, like 6 = 2 × 3).

First few: 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.

Circular Primes

A circular prime is a prime number where if you move its digits around (like rotating them), the new number is also prime. For example, if you have 197, you can rotate it to get 971 and 719, and all three are prime!

First few: 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.

Cousin Primes

Cousin primes are pairs of prime numbers that are exactly 4 apart.

First few pairs: (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).

Cuban Primes

Cuban primes are special primes that come from certain math formulas.

One type is found using the formula (x³ - y³) / (x - y) where x = y + 1.

* First few: 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.

Another type uses the formula (x³ - y³) / (x - y) where x = y + 2.

* First few: 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.

Cullen Primes

Cullen primes are numbers that follow the pattern: n × 2ⁿ + 1.

First few: 3, 393050634124102232869567034555427371542904833.

Dihedral Primes

Dihedral primes are primes that still look like prime numbers when you read them upside down or look at them in a seven-segment display (like on a digital clock).

First few: 2, 5, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081.

Eisenstein Primes

Eisenstein primes (without an imaginary part) are prime numbers that fit the pattern 3n - 1.

First few: 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.

Emirps

An emirp is a prime number that becomes a different prime number when its digits are reversed. The word "emirp" is "prime" spelled backward!

First few: 13 (reversed is 31, also prime), 17 (reversed is 71, also prime), 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.

Euclid Primes

Euclid primes are a type of primorial prime. They are found by multiplying the first few prime numbers together and then adding 1. For example, (2 × 3 × 5) + 1 = 31, which is prime.

First few: 3, 7, 31, 211, 2311, 200560490131.

Factorial Primes

Factorial primes are numbers that are either one less or one more than a factorial (a factorial of a number is that number multiplied by all the whole numbers smaller than it down to 1, like 5! = 5 × 4 × 3 × 2 × 1 = 120).

First few: 2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999.

Fermat Primes

Fermat primes are numbers of the form 2^2ⁿ + 1.

First few: 3, 5, 17, 257, 65537.
As of 2019, these are the only known Fermat primes. It's thought that there might not be any more!

Fibonacci Primes

Fibonacci primes are prime numbers that appear in the Fibonacci sequence. The Fibonacci sequence starts with 0 and 1, and each new number is found by adding the two numbers before it (0, 1, 1, 2, 3, 5, 8, 13, ...).

First few: 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917.

Fortunate Primes

Fortunate primes are prime numbers that are always larger than the product of the first n prime numbers plus one. For example, if you multiply the first few primes (2 × 3 × 5 = 30), the next prime after adding 1 (31) is a fortunate prime.

First few: 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.

Gaussian Primes

Gaussian primes are prime numbers that fit the pattern 4n + 3.

First few: 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.

Good Primes

Good primes are primes where their square is always larger than the product of any two primes that are equally far away from it in the list of primes.

First few: 5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307.

Happy Primes

Happy primes are prime numbers that are also happy numbers. A happy number is one where if you sum the squares of its digits, and then repeat the process with the new number, you eventually reach 1. For example, 7 is happy: 7² = 49, 4² + 9² = 16 + 81 = 97, 9² + 7² = 81 + 49 = 130, 1² + 3² + 0² = 1 + 9 + 0 = 10, 1² + 0² = 1.

First few: 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.

Home Primes

To find a home prime, you start with a number, write down its prime factors (the primes that multiply to make it), and then stick those factors together to make a new number. You keep doing this until you get a prime number. That final prime is the home prime.

First few: 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.

Isolated Primes

An isolated prime is a prime number that is not part of a twin prime pair (primes that are 2 apart, like 11 and 13) or a cousin prime pair (primes that are 4 apart, like 3 and 7).

First few: 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.

Leyland Primes

Leyland primes are numbers that follow the pattern x^y + y^x, where x and y are whole numbers greater than 1.

First few: 17, 593, 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193.

Long Primes

Long primes (also called full reptend primes) are primes p where the decimal expansion of 1/p has the longest possible repeating part (p - 1 digits).

First few: 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.

Lucas Primes

Lucas primes are prime numbers that appear in the Lucas sequence. This sequence starts with 2 and 1, and each new number is the sum of the two numbers before it (2, 1, 3, 4, 7, 11, ...).

First few: 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149.

Lucky Primes

Lucky primes are prime numbers that are also lucky numbers. Lucky numbers are found by a special sifting process, similar to how prime numbers are found.

First few: 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.

Mersenne Primes

Mersenne primes are very famous and are of the form 2ⁿ - 1. They are often the largest prime numbers discovered!

First few: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727.
As of 2018, there are 51 known Mersenne primes. The largest known prime number, M_82589933, is a Mersenne prime!

Double Mersenne Primes

These are a special kind of Mersenne prime, following the pattern 2^{2^p−1} - 1, where p is also a prime number.

First few: 7, 127, 2147483647, 170141183460469231731687303715884105727.

Mills Primes

Mills primes are generated by a special formula involving a constant called Mills' constant. This formula is guaranteed to produce prime numbers for all positive whole numbers.

First few: 2, 11, 1361, 2521008887, 16022236204009818131831320183.

Minimal Primes

Minimal primes are primes where you can't make a shorter prime by removing some of its digits.

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.

Palindromic Primes

Palindromic primes are prime numbers that read the same forwards and backwards, like a palindrome word (e.g., "madam").

First few: 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.

Pell Primes

Pell primes are prime numbers that appear in the Pell sequence. This sequence starts with 0 and 1, and each new number is found by multiplying the previous number by 2 and adding the number before that (0, 1, 2, 5, 12, 29, ...).

First few: 2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449.

Permutable Primes

Permutable primes are primes where any number you make by rearranging its digits is also a prime number.

First few: 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111 (19 ones), 11111111111111111111111 (23 ones).

Perrin Primes

Perrin primes are prime numbers found in the Perrin sequence. This sequence starts with 3, 0, 2, and each new number is the sum of the number two places before it and the number three places before it (3, 0, 2, 3, 2, 5, 5, 7, ...).

First few: 2, 3, 5, 7, 17, 29, 277, 367, 853, 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797.

Pierpont Primes

Pierpont primes are primes that can be written in the form 2^u3^v + 1, where u and v are whole numbers (0 or more).

First few: 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.

Primes of the form n⁴ + 1

These are prime numbers that can be found by taking a whole number n, raising it to the power of 4, and then adding 1.

First few: 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.

Primorial Primes

Primorial primes are numbers that are either one more or one less than a primorial. A primorial is like a factorial, but you only multiply prime numbers (e.g., 2# = 2, 3# = 2 × 3 = 6, 5# = 2 × 3 × 5 = 30).

First few: 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309.

Proth Primes

Proth primes are numbers that follow the pattern k × 2ⁿ + 1, where k is an odd number and smaller than 2ⁿ.

First few: 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.

Pythagorean Primes

Pythagorean primes are prime numbers that fit the pattern 4n + 1. These primes are special because they can be written as the sum of two perfect squares (e.g., 5 = 1² + 2²).

First few: 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.

Prime Quadruplets

Prime quadruplets are sets of four prime numbers that are very close together, in the form (p, p+2, p+6, p+8).

First few sets: (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).

Quartan Primes

Quartan primes are primes that can be written as the sum of two numbers, each raised to the power of 4 (x⁴ + y⁴).

First few: 2, 17, 97, 257, 337, 641, 881.

Ramanujan Primes

Ramanujan primes are special primes that help us understand how many primes there are up to a certain number.

First few: 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.

Repunit Primes

Repunit primes are prime numbers made up of only the digit 1.

First few: 11, 1111111111111111111 (19 ones), 11111111111111111111111 (23 ones).

Primes in Residue Classes

These are primes that fit specific patterns when you divide them by another number and look at the remainder. For example, primes of the form 4n + 1 always leave a remainder of 1 when divided by 4.

2n+1 (odd primes): 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53.
4n+1 (Pythagorean primes): 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137.
4n+3 (Gaussian primes): 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107.
10n+1 (primes ending in 1): 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, 271, 281.
10n+3 (primes ending in 3): 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263.
10n+7 (primes ending in 7): 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, 277.
10n+9 (primes ending in 9): 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, 349, 359.

Safe Primes

Safe primes are prime numbers p where (p - 1) / 2 is also a prime number.

First few: 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.

Self Primes

Self primes are prime numbers that cannot be made by adding a number to the sum of its own digits. For example, 10 is not a self number because 10 = 9 + (1+0).

First few: 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.

Smarandache–Wellin Primes

Smarandache–Wellin primes are formed by sticking together the first few prime numbers.

First few: 2, 23, 2357.
The next one is huge, with 355 digits!

Sophie Germain Primes

Sophie Germain primes are prime numbers p where 2p + 1 is also a prime number. These are named after the mathematician Sophie Germain.

First few: 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.

Super-primes

Super-primes are primes that appear at prime positions in the list of all prime numbers. For example, 3 is the 2nd prime (and 2 is prime), 5 is the 3rd prime (and 3 is prime), 11 is the 5th prime (and 5 is prime).

First few: 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.

Supersingular Primes

Supersingular primes are a special set of primes related to advanced math topics. There are only 15 of them!

The 15 supersingular primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71.

Thabit Primes

Thabit primes are numbers that follow the pattern 3 × 2ⁿ - 1.

First few: 2, 5, 11, 23, 47, 191, 383, 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407.
Related primes follow the pattern 3 × 2ⁿ + 1: 7, 13, 97, 193, 769, 12289, 786433, 3221225473, 206158430209, 6597069766657.

Prime Triplets

Prime triplets are sets of three prime numbers that are very close together. They come in two main forms: (p, p+2, p+6) or (p, p+4, p+6).

First few sets: (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).

Truncatable Primes

Truncatable primes are primes that stay prime even when you chop off digits from one end.

Left-Truncatable Primes

These primes remain prime when you remove the digit from the left side, one by one.

First few: 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.

Right-Truncatable Primes

These primes remain prime when you remove the digit from the right side, one by one.

First few: 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.

Two-Sided Truncatable Primes

These are primes that are both left-truncatable and right-truncatable.

There are exactly 15 two-sided primes: 2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397.

Twin Primes

Twin primes are pairs of prime numbers that are exactly 2 apart.

First few pairs: (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).

Unique Primes

Unique primes are primes where the length of the repeating part of their decimal fraction (like 1/7 = 0.142857142857...) is unique to that prime. No other prime will have the same length repeating part.

First few: 3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991.

Wagstaff Primes

Wagstaff primes are primes that follow the pattern (2ⁿ + 1) / 3.

First few: 3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243.

Weakly Prime Numbers

Weakly prime numbers are primes where if you change any single digit to any other digit, the new number will always be a composite number (not prime).

First few: 294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139.

Wilson Primes

Wilson primes are primes p where p² divides (p-1)! + 1. (Remember, p! means p × (p-1) × ... × 1).

First few: 5, 13, 563.
As of 2018, these are the only known Wilson primes.

Woodall Primes

Woodall primes are numbers that follow the pattern n × 2ⁿ - 1.

First few: 7, 23, 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319.