On .eth number mania and the rarity of numbers in Number Theory

When it comes to rarity among numbers, there are no superior alternatives than those numbers that are:

  • short (3/4/5 digits; more digits the worse) [already exploited :x::x::x:]
  • recognisably patterned (666, 999, 101, 010, 696, 776, 667, 69696 etc) [already exploited :x::x::x:]
  • culturally significant (420, 69, 13, 42, 111 etc) [already exploited :x::x::x:]
  • mathematically significant [not yet exploited :exclamation::star_struck:]
  • 0x prepended numbers [not yet exploited :exclamation:]

Some popular Prime number categories below (> 3 digits)

  • Rare/limited category

rare = fixed number of primes
limited = limited number known but more may exist

Fermat Primes [rare]

Only two known: 257, 65537!

Generalised Fermat Primes [rare]

Only 8 known: 101, 197, 257, 401, 577, 65537, 160001, 331777!

Dihedral Primes [rare]

101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081, 188011, 188801, 1008001, 1022201, 1028011, 1055501, 1058011, 1082801, 1085801, 1088081

Mersenne Prime Exponents [limited]

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, 82589933

Double Mersenne primes [rare]

127, 2147483647, 170141183460469231731687303715884105727

Minimal primes [rare]

409, 449, 499, 881, 991, 6469, 6949, 9001, 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049

Non-generous primes [limited]

40487, 6692367337

Permutable primes (non-repeating) [rare]

113, 131, 199, 311, 337, 373, 733, 919, 991

Two-sided [rare]

313, 317, 373, 797, 3137, 3797, 739397

  • Unlimited category

Balanced Primes

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 etc

Bell Primes

877, 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837 etc

Circular Primes

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 etc

Cousin Primes collection

(103, 107), (109, 113), (127, 131), (163, 167), (193, 197), (223, 227), (229, 233), (277, 281), (307, 311), (313, 317), (349, 353), (379, 383), (397, 401), (439, 443), (457, 461), (463,467), (487, 491), (499, 503), (613, 617), (643, 647), (673, 677), (739, 743), (757, 761), (769, 773), (823, 827), (853, 857), (859, 863), (877, 881), (883, 887), (907, 911), (937, 941), (967, 971) etc

Factorial Primes

719, 5039, 39916801, 479001599, 87178291199, 10888869450418352160768000001 etc

Fibonacci Primes

233, 1597, 28657, 514229, 433494437 etc

Fortunate primes

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 etc

Good primes

101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307 etc

Happy primes

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 etc

Lucky primes

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 etc

Palindromic primes

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 etc

Palindromic wing primes

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 etc

Sexy primes collection

(101, 107), (103, 109), (107, 113), (131, 137), (151, 157), (157, 163), (167, 173), (173, 179), (191, 197), (193, 199) etc

Strobogrammatic primes

101, 181, 619, 16091, 18181, 19861, 61819, 116911, 119611, 160091, 169691, 191161, 196961, 686989, 688889 etc

Prime triplets collection

(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) etc

Twin primes collection

(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) etc

Happy registering!