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
] - recognisably patterned (666, 999, 101, 010, 696, 776, 667, 69696 etc) [already exploited ]
- culturally significant (420, 69, 13, 42, 111 etc) [already exploited ]
- mathematically significant [not yet exploited
] - 0x prepended numbers [not yet exploited ]
Some popular Prime number categories below (> 3 digits)
rare = fixed number of primes
limited = limited number known but more may exist
Fermat Primes [rare]
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
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
(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
This is an epic post, thanks for sharing!
The only number I will ever mint is the largest prime number under maximum gas per block (~1.5-2 ETH to max out a block at current gas). That is an unsolved problem (finding the largest prime number of N digits where N ~ 1,000,000 digits or more) in mathematics in general.