THE TOP TEN PRIME NUMBERS

(a catalogue of primal configurations) from the unpublished collections of R. Ondrejka

assisted by C. Caldwell and H. Dubner May 05, 2001

Table of Contents

Introduction  4
An Unalphabetical Glossary  5
Table 1:  The TOP TEN Prime Numbers  8
Table 1.A:  The TOP TEN Mersenne Prime Numbers  9
Table 1.B:  The TOP TEN non-Mersenne prime Numbers  10
Table 2:  The TOP TEN Prime Factors of Fermat Numbers  11
Table 2.A  The Five Known Fermat Prime Numbers  11
Table 2 B : The TOP TEN Generalized Fermat Prime Numbers  12
Table 3.A:  The TOP TEN Cullen Primes: C_{n} = n · 2^{n} + 1  13
Table 3.B:  The TOP TEN Woodall Primes: W_{n} = n · 2^{n} − 1  14
Table 4:  The TOP TEN Known Largest Primes with Total Digits Prime  15
Table 5:  The TOP TEN Naughtiest Prime Numbers  16
Table 6 :  The TOP TEN Almost-All-Even-Digits Prime Numbers  17
Table 7:  The TOP TEN Quasiall-Even-Digits Prime Numbers  18
Table 8:  The TOP TEN All Odd Digits Prime Numbers  19
Table 9:  The TOP TEN Primes with Long Repdigit Strings  20
Table 9 A:  The TOP TEN Leading Repdigit Prime Numbers  21
Table 9.B:  The TOP TEN Internal Repdigit (0) Prime Numbers  22
Table 9 B.1: The TOP TEN Internal Repdigit (nonzero) Primes  23
Table 9.C:  The TOP TEN Ending Repdigit(9) Primes  24
Table 10:  The TOP TEN Palindromic Prime Numbers  25
Table 10.A: The TOP TEN Tetradic Prime Numbers  26
Table 10.B: The TOP TEN Triadic Prime Numbers  27
Table 10.C: The TOP TEN Quasi-Even-Digits Palindromic Prime Numbers  28
Table 10.D: The TOP TEN Zero-free Palindromic Primes  29
Table 10.E: The TOP TEN Near-Repdigit Palprimes  30
Table 10.F: The TOP TEN All-Odd Digits Palprimes  31
Table 10.G: The Top Beastly Palindromic Prime Numbers  32
Table 10.G.1: The TOP TEN Generalized Beastly Palindromic Primes  33
Table 10.H: The TOP TEN Pandigital, Palindromic Prime Numbers  34
Table 10.I: The TOP TEN Alternate-Digit Palprimes 35
Table 10.I.1: The TOP TEN Primes with Alternating Unholey/Holey Digits  36
Table 10.I.2: The TOP TEN Primes with Alternating Straight/Curved Digits  37
Table 10.I.3: The TOP TEN Primes with Alternating Prime/Composite Digits  38
Table 10.J.: The TOP TEN Palprimes with Composite Digits & The TOP TEN Holey Palprimes 39
Table 10.K: The TOP TEN Palprimes with Curved Digits  40
Table 10.L.: The TOP TEN Unholey Palprimes  41
Table 10 M.: The TOP TEN Invertible Palprime Pairs  42
Table 10 N.: The TOP TEN Smoothly Undulating Palprimes  43
Table 10.O: The TOP TEN Palprimes with Prime Digits  44
Table 10.P: The TOP TEN Palindromic Quasi-Repdigit Prime Numbers  45
Table 10.P.1: The TOP TEN P^{latea}u Prime Numbers  46
Table 10.P.2: The TOP TEN D_{epressio}n Prime Numbers  47
Table 10.Q: The TOP TEN Palprimes with Straight Digits  48
Table 10.R: The Five Repunit Primes Known  48
Table 10.S:  The TOP TEN Known Palindromic Primes in Arithmetic Progression  49
Table 11:  The TOP TEN Primes with Square Digits  51
Table 12:  The TOP TEN Prime Numbers with Cube Digits  52
Table 13:  The TOP TEN Anti-Yarborough Primes with 1’s and 0’s Digits  53
Table 14:  The TOP TEN Yarborough Prime Numbers; The TOP TEN Zero-free Primes  53

2

Table 15:  The TOP TEN Prime Twins  54
Table 15.A: The TOP TEN Prime Triplets  55
Table 15.B: The Top k-tuplets of Primes  56
Table 16.A: The TOP TEN Near Repdigit Prime Numbers AB_{n}  57
Table 16.B: The TOP TEN Near-Repdigit Prime Numbers A_{n−k−1} BA_{k}  58
Table 16.C: The TOP TEN Near Repdigit Prime Numbers A_{n}B  59
Table 16.D: The TOP TEN Near-Repunit Prime Numbers  60
Table 17:  The TOP TEN Quasi-Repdigit Prime Numbers  61
Table 18:  The TOP TEN Factorial Prime Numbers  62
Table 18.A: The TOP TEN Factorial-Plus-One Primes  63
Table 18.B: The TOP TEN Factorial-Minus-One Primes  64
Table 19:  The TOP TEN Primes of Alternating Sums of Factorials  65
Table 20:  The TOP TEN Primorial Prime Numbers  66
Table 20.A: The TOP TEN Primorial-Plus-One Primes  67
Table 20.B: The TOP TEN Primorial-Minus-One Primes  68
Table 21: The TOP TEN Multifactorial Prime Numbers 69
Table 21.A: The TOP TEN Double Factorial Prime Numbers  70
Table 21.B: The TOP TEN Triple Factorial Prime Numbers  71
Table 21.C: The TOP TEN Quadruple Factorial Prime Numbers  72
Table 21.D: The TOP TEN Quintuple Factorial Prime Numbers  73
Table 21.E: The TOP TEN Sextuple Factorial Prime Numbers  74
Table 21.F: The TOP TEN Septuple Factorial Prime Numbers  75
Table 22:  The TOP TEN Primes w. Composite Digits & the TOP TEN Holey Primes 76
Table 23:  The TOP TEN Undulating Prime Numbers  77
Table 24:  The TOP TEN Primes with Curved Digits  78
Table 25:  The TOP TEN Sophie Germain Primes  79
Table 26:  The TOP TEN Anti-Palindromic Prime Numbers  80
Table 27:  The TOP TEN Generalized Repunit Primes  81
Table 28:  The TOP TEN Strobogrammatic Primes (nonpalindromic)  82
Table 29:  The TOP TEN Beastly Primes (nonpalindromic)  83
Table 30:  The TOP TEN Sub_{script} Prime Numbers  84
Table 31:  The TOP TEN Unholey Primes  85
Table 32:  The TOP TEN Prime Numbers with Prime Digits  86
Table 33:  The TOP TEN Lucas Prime Numbers  87
Table 34:  The TOP TEN Prime Fibonacci Numbers  88
Table 35:  The TOP TEN Countdown Prime Numbers  89
Table 36:  The TOP TEN Reversible Primes (nonpalindromic)  90
Table 37:  The TOP TEN Primes with Straight Digits  91
Table 38:  The TOP TEN Primes with Largest Unique Periods  92
Table 39:  The TOP TEN Absolute Prime Numbers  93
Talbe 40:  The TOP TEN Consecutive Primes in Arithmetic Progression  94
Table 41:  The TOP TEN Known Factors of Googolplex Plus One  95
Table 42:  Ten Types of “Rare” Prime Numbers  96

Introduction

The infinity of prime numbers (or primes) may be categorizied in many ways, as will be seen in the TOP TEN tabular listings or charts. One may divide the primes into two groups: the Mersenne and non-Mersenne types. The former, named after the French friar-mathematician Marin Mersenne (1588-1648), are rather special, as they are generally the world’s largest known. Also, they are directly linked with even perfect numbers: [2^{p−1}. (2^{p} − 1)]; those rare numbers whose \underline{proper} divisors sum to the above expression. For completeness, all known Mersenne primes (2^{P} −1) can be found in table 1.A, including the latest and largest one.

The non-Mersenne primes, infinite in number, may be subdivided into palindromic and non-palimdromic classes. The palindromic prime numbers (or palprimes) are thought to be infinite in number, but this is only a conjecture. Of the 20 palprime tables listed, six will be mentioned: (1) the rare repunit primes, R_{n}, which are a small subclass of the tetradics; (2) the tetradic or 4-way primes: those palprimes which are unchanged when turned upside down or reflected in a mirror; (3) the triadic or 3-way primes; (4) the near repdigit, which are some of the prettiest palprimes, having all like digits except for the center digit; (5) Q-E-D palprimes, those whose digits are all \underline{even}, except for the end, odd digits; (6) pandigital palprimes are those having all ten (0 to 9) digits represented.

The class of nonpalindromic primes are an infinite and diverse lot. Listing of “The TOP TEN” primorials, factorials, and multifactorials are found in the following pages. These primes are: products of consecutive primes, plus or minus unity; products of the first N consecutive integers, plus or minus one; and generalizations of the factorials, respectively. The famous Fermat primes (2^{2^{n}} + 1) and generalizations of the Fermat primes are charted, along with listings of Cullen primes and Sophie Germain primes. Some primes on a list (like Fibonacci or Cullen) exceed ten in number, and if all are known, are included for completeness; a few lists (like Fermat or repunits), are also included, though lacking the requisite number (10). As in all the lists, the number 1 prime will include its discoverer(s) and date.

A set of primes having many zeros (thousands), or primes with none are reported; as well as primes with long repdigit strings; most leading and ending strings of thousands of like digits. There are tables of “odd” digit primes \underline{and} odd digit primes, as well as primes having \underline{almost} all even digits. The digits of primes may be characterized in other ways: primes with all \underline{prime} digits or all composite digits; all “straight” digits or all “curved” digits, and digits with “holes”; even primes having all digits “square” or all “cube”. Tables of absolute primes, alternate-digit primes and antipalindromic primes, among others are here for the asking. Among the 80 \underline{odd} tables or charts are found: the largest known TOP TEN prime twin pairs; beastly and unholey primes (two entirely different species); undulating primes, unique-period primes, etc.

On the last page is a table of rare primes ranked by rarity. Should anyone having a fast computer, decide to set a new record by finding a larger, rare prime, one should first tackle the “easier” types, like: “B”, “C”, “G”, or “J”.

A small glossary follows for those who may be unfamiliar with some of the nomenclature above, or technical terms in the prime listings.

An Unalphabetical Glossary

A Mersenne Prime, generally the worlds largest, is of the form M_{p} = 2^{p} − 1, p always prime. A naught-y or naught-iest prime is one having a very high percentage of naughts or zeros. A near-repdigit prime is one with all like or repeated digits, but one. A quasi-repdigit prime is an all repeating digit prime, except for two digits. Q-E-D prime: a quasiall-even-digits prime-a prime with even digits and two odd digits. An almost-all-even-digits prime: a prime with even digits and one odd, right-most digit. A palindromic prime (or palprime), is the same, whether read from left to right or vice versa. A tetradic prime is a 4-way prime: palindromic, as well as the same upside down and mirror reflected. A triadic prime is a 3-way prime: having up-down or vertical mirror symmetry, as well as palindromicity. A beastly prime is a palindrome with 666 in the center, 0’s surrounding these digits, and 1 or 7 at the end. A non-palindromic beastly prime will start with 666, followed by 0’s, and either a 1 or 7 at the right end. Pandigital means all 10 digits (0 to 9); almost pandigital means 9 digits (usually zero is the missing digit). Almost-equi-pandigital means all digits are equal in number, except for one particular digit. Prime digit primes only have the digits: 2, 3, 5, or 7; composite digit primes have only: 4, 6, 8, or 9. Holey primes are primes having digits with holes, like the composite digits primes, and can include the zero. Wholly, holey primes are primes, all of their digits have holes; unholey primes do not have holes in digits. A repunit prime, R_{n} = (10^{n} − 1)/9, is a prime with all repeated 1’s. A near-repunit prime has almost all repeated units, except one. An alternate-digit prime has alternating odd and even digits. In an undulating prime number, the neighboring digits are consistently greater or less than the digits adjacent to them. In a smoothly undulating prime (palindromic), only two types of digits are involved. A Yarborough prime can have any number of the digits: 2, 3, 4, 5, 6, 7, 8, or 9, \underline{only}.

A depression prime is a palprime having all interior digits \underline{repeating}, and smaller than its two end digits. A plateau prime is a palprime having all interior digits \underline{repeating}, and larger than its two end digits. A factorial prime is the product of the first n consecutive integers, plus or minus unity. A primorial prime is the product of the first n consecutive primes, plus or minus unity. Multifactorial primes N!_{k} ± 1, are generalizations of the factorial prime:

double factorial primes (N! ! ± 1) = N · (N − 2)(N − 4)(N − 6) ⋯ ± 1; triple factorial primes (N! ! ± 1) = N · (N − 3)(N − 6)(N − 9) …. , etc.

An odd digit prime has a single “odd” digit in its decimal representation; an all odd digit prime has all its digits odd. A Fermat prime is of the form 2^{2^{N}} + 1; a generalized Fermat prime is of the form b^{2^{N}} + 1. Primes with curved digits are composed only of 0’s, 3’s, 6’s, 8’s, or 9’s; primes with straight digits only, include 1’s, 4’s, or 7’s. An anti-Yarborough prime can have any number of 1’s and 0’s only. An antipalindromic prime must have a total \underline{even} number of digits; and the digits in the first half of the prime \underline{must} differ from the corresponding digits of the second half, resulting in a coincidence ratio of \underline{zero}. A Cullen prime is of the form: C_{n} = n · 2^{n} + 1; A Woodall prime is of the form: W_{n} = n · 2^{n} − 1. Sub_{script}primes are so called because they are usually expressed in their subscriptal notation. A Sophie Germain prime is an odd prime p for which 2p + 1 is also prime. A unique-period prime (P) is one, whose reciprocal (1/p), is a period not shared by any other prime. An absolute prime is one that remains a prime, for all permutations of its digits. Generalized repunit primes are primes of the form (b^{n} − 1)/(b − 1) b not equal to 2 or 10. A strobogrammatic prime remains unchanged when turned upside down or rotated 180^{0}). A generalized beastly prime is a palindrome \underline{not} limited to only zeros surrounding the infamous “666.” A “k-tuplet” is a sequence of k consecutive primes: q_{1} q_{2} … q_{k}, with q_{k} − q_{1} as small as possible. Prime twins (p and p + 2) are the smallest of these sequences; followed by prime triplets, quadruplets, etc. A countdown prime is one whose leading digits descend: 10 or 9 8 7 6 5 4 3 2 1.

Lucas numbers (L_{n}) unlike their famous Fibonacci cousins (F_{n}), are derived initially from the sequence 2, 1, 3, 4, 7, 11, 18, 29, 47. An Invertible prime number is one that, when turned upside down (revolved 180^{0}), results in a new different prime. Digits like 0, 1, 6, 8, or 9, must be only used. A Nonpalindromic reversible prime (also known as an emirp), produces a different prime, when all its digits are reversed; .e.g. 13 and 31. Googolplex, a number impossible to represent fully in decimal, is equal to 10^{10^{10}}, or ten to the googol power, i.e. one followed by a zeros [10^{100}zeros]. A Googol is a number of 101 digits, or more precisely, one followed by 100 zeros. Subscriptal digits are sometimes used to show repeated decimal digits; e.g. 13_{4} = 13333; (23)_{3} = 232323. A Titanic Prime is a number having one thousand or more digits; a Gigantic Prime has ten thousand or more digits; a Megaprime is a prime number having one million or more digits.

Table 1. The TOP TEN Prime Numbers

[M_{p} = 2^{p} − 1; k · 2^{N} ± 1]

Digits

1. M_{6972593} = 2^{6972593} − 1  2098960 [N.Hajratwala, etal:1999]
*2. M_{3021377} = 2^{3021377} − 1  909526
*3. M_{2976221} = 2^{2976221} − 1  895932
*4. M_{1398269} = 2^{1398269} − 1  420921
*5. M_{1257787} = 2^{1257787} − 1  378632
**6. 48594^{65536} + 1  307140
*7. M_{859433} = 2^{859433} − 1  258716
*8. M_{756839} = 2^{756839} − 1  227832
***9. 667071 · 2^{667071} − 1  200815
**10. 1041870^{32768} + 1  197192

*  Former Mersenne world record holder
**  Generalized Fermat prime number
***  Woodall prime number

Table 1. A: The TOP TEN Mersenne Prime Numbers

[M_{p} = 2^{p} − 1]

Digits

1. M_{6972593} = 2^{6972593} − 1 2098960 [N.Hajratwala, etal;1999]
2. M_{3021377} = 2^{3021377} − 1 909526
3. M_{2976221} = 2^{2976221} − 1 895932
4. M_{1398269} = 2^{1398269} − 1 420921
5. M_{1257787} = 2^{1257787} − 1 378632
6. M_{859433} = 2^{859433} − 1  258716
7. M_{756839} = 2^{756839} − 1  227832
8. M_{216091} = 2^{216091} − 1  65050
9. M_{132049} = 2^{132049} − 1  39751
10. M_{110503} = 2^{110503} − 1  33265

[and the rest]

D  D  D
11. M_{86243} 25962  20. M_{4253} 1281  29. M_{89} 27
12. M_{44497} 13395  21. M_{3217} 969  30. M_{61} 19
13. M_{23209} 6987  22. M_{2281} 687  31. M_{31} 10
14. M_{21701} 6533  23. M_{2203} 664  32. M_{19} 6
15. M_{19937} 6002  24. M_{1279} 386  33. M_{17} 6
16. M_{11213} 3376  25. M_{607} 183  34. M_{13} 4
17. M_{9941} 2993  26. M_{521} 157  35. M_{7} 3
18. M_{9689} 2917  27. M_{127}  39  36. M_{5} 2
19. M_{4423} 1332  28. M_{107}  33  37. M_{3} 1
38. M_{2} 1

Table 1. B: The TOP TEN Non-Mersenne Prime Numbers

Digits

*1. 48954^{65536} + 1  307140 [S.Scott & Y.Gallot:2000]
**2. 667071 · 2^{667071} − 1  200815
*3. 1041870^{32768} + 1  197192
*4. 999236^{32768} + 1  196598
*5. 524552^{32768} + 1  187427
*6. 167176^{32768} + 1  171153
7. 169719 · 2^{557557} + 1  167487
8. 302627325 · 2^{530101} + 1 159585
9. 43541 · 2^{507098} − 1  152657
10. 144643 · 2^{498079} − 1  149942
* Generalized Fermat prime
** Woodall prime

Table 2: The TOP TEN Prime Factors of Fermat Numbers F_{m} = 2^{2^{m}} + 1 [k · 2^{n} + 1] Fermat

Digits exp (m)
1. 3 · 2^{382449} + 1 115130 382447 [J. Cosgrove & Y. Gallot:1998]
2. 3 · 2^{303093} + 1  91241 303088
3. 3 · 2^{213321} + 1  64217 213319
4. 3 · 2^{157169} + 1  47314 157167
5. 57 · 2^{146223} + 1 44020 146221
6. 5 · 2^{125413} + 1  37754 125410
7. 13 · 2^{114296} + 1 34408 114293
8. 39 · 2^{113549} + 1 34184 113547
9. 7 · 2^{95330} + 1  28699 95328
10. 21 · 2^{94801} + 1 28540 94798

Table 2. A: The Five Known Fermat Prime Numbers F_{m} = 2^{2^{m}} + 1

Actual* Primes

1. F_{4} = 2^{2^{4}} + 1 = 65537 [P. Fermat: 1640]
2. F_{3} = 2^{2^{3}} + 1 =  257
3. F_{2} = 2^{2^{2}} + 1 =  17
4. F_{2} = 2^{2^{1}} + 1 -  5
5. F_{0} = 2^{2^{0}} + 1 =  3

* Actual Primes; the next 26 Fermat numbers, F_{5} to F_{30} are composite Table 2. B: The TOP TEN Generalized Fermat Prime Numbers

[P = b^{2^{n}} + 1] Digits

1. 48594^{2^{16}} + 1  307140 [S. Scott & Y. Gallot: 2000]
2. 1041870^{2^{15}} + 1 197192
3. 999236^{2^{15}} + 1 196598
4. 524552^{2^{15}} + 1 187427
5. 167176^{2^{15}} + 1 171153
6. 840796^{2^{14}} + 1  97071
7. 704930^{2^{14}} + 1  95817
8. 656210^{2^{14}} + 1  95307
9. 641762^{2^{14}} + 1  95149
10. 638980^{2^{14}} + 1  95118

Table 3. A: The TOP TEN Cullen Primes C_{n} = n · 2^{n} + 1

C_{n}  Digits
1. 481899 · 2^{481899} + 1 145072 [M.Morii & Y.Gallot:1998]
2. 361275 · 2^{361275} + 1 108761
3. 262419 · 2^{262419} + 1 79002
4. 90825 · 2^{90825} + 1  27347
5. 59656 · 2^{59656} + 1  17964
6. 32469 · 2^{32469} + 1  9779
7. 32292 · 2^{32292} + 1  9726
8. 18496 · 2^{18496} + 1  5573
9. 6611 · 2^{6611} + 1  1994
10. 5795 · 2^{5795} + 1  1749

[and the rest]

11. 4713 · 2^{4713} + 1  1423
12. 141 · 2^{141} + 1  45
13. 1 · 2^{1} + 1  1

Table 3. B: The TOP TEN Woodall Primes W_{n} = n · 2^{n} − 1

W_{n}  Digits
1. 667071 · 2^{667071} − 1 200815 [M.Toplic & Y.Gallot:2000]
2. 151023 · 2^{151023} − 1 45468
3. 143018 · 2^{143018} − 1 43058
4. 98726 · 2^{98726} − 1  29725
5. 23005 · 2^{23005} − 1  6930
6. 22971 · 2^{22971} − 1  6920
7. 18885 · 2^{18885} − 1  5690
8. 15822 · 2^{15822} − 1  4768
8. 12379 · 2^{12379} − 1  3731
10. 9531 · 2^{9531} − 1  2874

[ and the rest]

W_{n}  D  W_{n}  D
11. 7755 · 2^{7755} − 1 2339  20. 123 · 2^{123} − 1 40
12. 5312 · 2^{5312} − 1 1603  21. 115 · 2^{115} − 1 37
13. 822 · 2^{822} − 1  251  22. 81 · 2^{81} − 1 27
14. 751 · 2^{751} − 1  229  23. 75 · 2^{75} − 1 25
15. 512 · 2^{512} − 1  157  24. 30 · 2^{30} − 1 11
16. 462 · 2^{462} − 1  142  25. 6 · 2^{6} − 1  3
17. 384 · 2^{384} − 1  119  26. 3 · 2^{3} − 1  2
18. 362 · 2^{362} − 1  112  27. 2 · 2^{2} − 1  1
19. 249 · 2^{249} − 1  78

Table 4: The TOP TEN Known Largest Primes with Total Digits Prime

[k · 2^{n} ± 1]

Total Digits

1. 43541 · 2^{507098} − 1  152657 [R. Ballinger & Y. Gallot: 2000]
* 2. 361275 · 2^{361275} + 1  108761
3. 115947 · 2^{350003} + 1  105367
4. 189453 · 2^{324103} − 1  97571
** 5. 422666^{16384} + 1  92177
6. 144817 · 2^{258857} − 1  77929
7. 15809 · 2^{256640} − 1  77261
8. 892451707 · 2^{239848} + 1  72211
9. 73 · 2^{227334} + 1  68437
*** 10. 3 · 2^{213321} + 1  64217

* Cullen prime
** Generalized Fermat prime
*** Fermat factor of 2^{213319} + 1

Table 5: The TOP TEN Naughtiest Prime Numbers [ Naught/cifre/zero/egg/oh/0 ]

Total

Naughts Naught% Digits
* 1. 105594 · 10^{105994} + 1  105994  99.994 106000 [Loeh & Gallot: 2000]
2. 193 · 10^{69004} + 1  69003  99.994 69007
3. 49521 · 10^{49521} + 1  49520  99.988 49526
** 4. 10^{35352} + 2049402 · 10^{17673} + 1  35346  99.980 35353
5. 127 · 10^{31000} + 1  30999  99.987 31003
** 6. 10^{30802} + 1110111 · 10^{15398} + 1  30795  99.974 30803
7. 3 · 10^{27720} + 1  27719  99.993 27721
8. 9964227 · 10^{21244} + 1  21243  99.96 21251
9. 247 · 10^{20006} + 1  20005  99.98 20009
10. 10^{20000}10^{19536} + 1  19535  97.68 20000
* Most consecutive zeros; highest zero %; more zeros than any known prime except M_{38}
** Palindrome

Table 6: The TOP TEN Almost-All-Even-Digits Prime Numbers **

[K · 10^{N} + 1; D · R(N) · 10^{K} + 1, D = 2, 4, 6, 8]
Digits E-D%
1. 8 · R(12600) · 10^{3705} + 1  16305 99.994 [H. Dubner:1997]
*2. 666 · 10^{14020} + 1  14023 99.993
3. 80602 · 10^{14013} + 1  14018 99.993
4. 4 · R(10200) · 10^{2894} + 1  13094 99.992
5. 2 · R(10200) · 10^{2396} + 1  12596 99.992
6. 6 · R(10200) · 10^{2057} + 1  12257 99.992
7. 8 · R(10080) · 10^{1003} + 1  11083 99.991
8. 4 · R(9240) · 10^{151} + 1  9391 99.989
*9. 666 · 10^{9198} + 1  9201 99.989
10. 4 · R(7560) · 10^{1023} + 1  8583 99.988

* Beastly prime (non palindromic)
** All primes discovered by H. Dubner

Table 7: The TOP TEN Quasiall-Even-Digits Prime Numbers

[Q − E − D Primes; 0, 2, 4, 6, 8]
Digits E-D%
**1. (10_{14285}80_{14285}1)  28573 99.993 [D. Heuer:2001]
* 2. 30_{27719}1  27721 99.993
3. 2470_{20005}1  20009 99.990
4. 10_{13326}20840_{6658}1  20000 99.990
** 5. (10_{7771}42606240_{7771}1)  15551 99.98
*** 6. 261840_{12090}1  12096 99.98
* 7. 30_{10452}1  10454 99.98
8. 362640_{8061}1  8067 99.98
** 9. (10_{3525}22202220_{3525}1)  7059 99.97
**** 10. (10_{2864}6_{15}0_{2864}1)  5745 99.97

* Quasi-repdigit prime
** Palindrome
*** All even digits represented
**** Generalized beastly palindrome

Table 8: The TOP TEN All Odd Digits Prime Numbers

[K · 10^{N} − 1]

Digits

*1. 10^{50103} − 4 · 10^{50097} − 1  or 9_{5}59_{50097}  50103 [P. Carmody:2000]
* 2. 10^{25000} − 4 · 10^{18479} − 1  or 9_{6521}59_{18479} 25001
* 3. 10^{19999} − 2 · 10^{18038} − 1  or 9_{1960}79_{10838} 19999
** 4. 2 · 10^{19233} − 1  or 19_{19233}  19234
** 5. 6 · 10^{18668} − 1  or 59_{18668}  18669
** 6. 2 · 10^{15749} − 1  or 19_{15749}  15750
** 7. 8 · 10^{11336} − 1  or 79_{11336}  11337
*** 8. 10 · 159795 · R(10080)/R(6) + 1 or (159795)_{1680}1 10081
** 9. 2 · 10^{7517} − 1  or 19_{7517}  7518
** 10. 2 · 10^{5969} − 1  or 19_{5969}  5970

* Near-repdigit prime
** Near repdigit string prime
*** Palindrome

Table 9: The Ten Top Primes with Long Repdigit Strings

[K · 10^{N} ± 1]
Repdigit String % Total
String  Digits
* 1. 105994 · 10^{105994} + 1  105993 99.993  106000 [G. Loeh & Y. Gallot: 2000]
2. 193 · 10^{69004} + 1  69003 99.994  69007
**3. 10^{50103} − 4 · 10^{50097} − 1  50097 99.998  50103
4. 49521 · 10^{49521} + 1  49520 99.988  49526
*** 5. 9 · 10^{48051} − 1  48051 99.998  48052
***6. 9 · 10^{41475} − 1  41475 99.998  41476
**** 7. 3 · 10^{33058} − 1  33058 99.997  33059
8. 127 · 10^{31000} + 1  30999 99.987  31003
*****9. 3 · 10^{27720} + 1  27719 99.993  27721
**** 10. 3 · 10^{26044} − 1  26044 99.996  26045

*  Most consecutive 0’s
**  Most consecutive ending 9’s
***  Highest string %
**** Near-repdigit prime
***** Q-E-D prime

Table 9. A: The TOP TEN Leading Repdigit Prime Numbers

[D · R(n) · 10^{k} + 1]
Digit  Repdigits String % Total

Digits

1. R(10080) · 10^{2136} + 1 or 1_{10080}0_{2135}1  10080 82.51  12216 [1996]
2. 2R(10200) · 10^{2396} + 1 or 2_{10200}0_{2395}1  10200 80.98  12596 [1996]
3. 3R(10080) · 10^{2286} + 1 or 3_{10080}0_{2285}1  10080 81.51  12366 [1996]
4. 4R(10200) · 10^{2894} + 1 or 4_{10200}0_{2893}1  10200 77.90  13094 [1996]
* 5. 5R(12600) · 10^{68} + 1 or 5_{12600}0_{67}1  12600 99.46  12668 [1997]
6. 6R(10200) · 10^{2057} + 1 or 6_{10200}0_{2056}1  10200 83.22  12257 [1996]
7. 7R(12600) · 10^{381} + 1 or 7_{12600}0_{380}1  12600 97.06  12981 [1996]
8. 8R(12600) · 10^{3705} + 1 or 8_{12600}0_{3704}1  12600 77.28  16305 [1997]
**9. 10^{38500} − 10^{18168} − 1 or 9_{20332}89_{18168}  20332 52.81  38501 [2000
*** 9. 10^{30000} − 10^{13560} − 1 or 9_{16440}89_{13560}  16440 54.80  30001 [2000]

*  Highest string percentage
** Most leading repdigit
*** Discovered by P. Underwood; all other primes by H. Dubner

Table 9. B: The TOP TEN Internal Repdigit (0) Prime Numbers

[K · 10^{N} + 1]
Repdigit  String % Total
String  Digits
1. 105994 · 10^{105994} + 1  105993  99.993  106000 [G. Loeh & Y. Gallot:2000]
*2. 193 · 10^{69004} + 1  69003  99.994  69007
3. 49521 · 10^{49521} + 1  49520  99.988  49526
4. 127 · 10^{31000} + 1  30999  99.987  31003
5. 3 · 10^{27720} + 1  27719  99.993  27721
6. 9964227 · 10^{21244} + 1  21243  99.96  21251
7. 247 · 10^{20006} + 1  20005  99.98  20009
8. 10^{20000} − 10^{19536} + 1  19535  97.67  20001
**9. 10^{35352} + 204\underline{9}402 · 10^{17673} + 1  17672  49.99  35353
10. 10^{16201} + 37  16199  99.98  16202

* Highest string %
** Palindrome

Table 9. B. 1: The TOP TEN Internal Repdigit (nonzero) Primes **

Digit  Repdigits String % Total Digits
1 16671_{10075}094441  10075 99.90  10085
2 113882_{10075}108341  10075 99.89  10086
2 223072_{10074}1999151  10074 99.88  10086
3 643_{7557}2691  7557 99.92  7563
4 12896734_{5033}31547711  5033 99.70  5048
*5 285_{10078}271  10078 99.95  10083
6 26_{6995}39_{54}1_{6996}  6995 49.80  14047
7 35047_{10076}42731  10076 99.91  10085
8 13965168_{5033}74923721  5033 99.70  5048
9 17309_{10076}82691  10076 99.91  10085

* Most internal repdigits/highest string %
** All discovered by H. Dubner (1997)

Table 9. C: The TOP TEN Ending Repdigit (9) Primes **

[AB_{n}; K · 10^{n} − 1]
Repdigit String % Total
String  Digits
1 10^{50103} − 4 · 10^{50097} − 1  50097 99.988  50103 [P. Carmody:2000]
*2. 9 · 10^{48051} − 1  48051 99.988  48052
*3. 9 · 10^{41475} − 1  41475 99.998  41476
*4. 3 · 10^{33058} − 1  33058 99.997  33059
*5. 3 · 10^{26044} − 1  26044 99.996  26045
6. 10^{30005} − 10^{23906} − 1  23906 79.67  30006
7. 10^{30007} − 10^{22717} − 1  22717 75.70  30008
8. 10^{25000} − 7 · 10^{22632} − 1  22632 90.52  25001
9. 10^{30006} − 10^{21425} − 1  21425 71.40  30007
10. 10^{30002} − 10^{21020} − 1  21020 70.06  30003

* Near-repdigit string prime
** Plus three other ending repdigit (1, 3, 7) primes:

Repdigit  Repdigit String% Total
String  Digits

(1)  26_{6995}39_{54}1_{6996}  6996  49.80 14047
(3)  13_{5216}19_{2300}3_{5217}  5217  40.97 12735
(7)  2668319_{5694}7331680_{2299}17_{5700} 5700  41.59 13706

Table 10: The TOP TEN Palindromic Prime Numbers

[10^{A} + K · 10^{B} + 1]

Digits

1. 10^{35352} + 2049402 · 10^{17673} + 1 or (10_{17672}20494020_{1762}1) 35353 [H.Dubner :1999]
* 2. 10^{30802} + 1110111 · 10^{15398} + 1 or (10_{15397}11101110_{15397}1) 30803
* 3. 10^{28572} + 8 · 10^{14286} + 1  or (10_{14285}80_{14285}1)  28573
4. 10^{19390} + 4300034 · 10^{9692} + 1 or (10_{9691}43000340_{9691}1) 19391
5. 10^{16650} + 53735 · 10^{8323} + 1  or (10_{8322}537350_{8322}1)  16651
6. 10^{16360} + 3644463 · 10^{8177} + 1 or (10_{8176}36444630_{8176}1) 16361
** 7. 10^{15640} + 3 · 10^{7820} + 1  or (10_{7819}30_{7819}1)  15641
8. 10^{15550} + 7410147 · 10^{7772} + 1 or (10_{7771}74101470_{7771}1) 15551
8. 10^{15550} + 7105017 · 10^{7772} + 1 or (10_{7771}71050170_{7771}1) 15551
*** 8. 10^{15550} + 4260624 · 10^{7772} + 1 or (10_{7771}42606240_{7771}1) 15551
8. 10^{15550} + 3698963 · 10^{7772} + 1 or (10_{7771}36989630_{7771}1) 15551

*  Tetradic or 4-way prime
** Triadic or 3-way prime
*** Q-E-D palprime

Table 10. A: The TOP TEN Tetradic Prime Numbers [4-way Palprimes] [ 0, 1, 8 ]

Digits

* 1. 10^{30802} + 1110111 · 10^{15398} + 1  or (10_{15397}1110111_{15397}1)  30803 [H.D. :1999]
** 2. 10^{28572} + 8 · 10^{14286} + 1  or (10_{14285}80_{14285}1)  28573
* 3. 10 · 110101 · R(10080)/R(6) + 1  or (110101)_{1680}1  10081
4. 10^{6906} + 88111881818811188 · 10^{3445} + 1 or (10_{3444}881118818188111880_{3444}1) 6907
* 5. 10^{4840} + 111111111 · 10^{2416} + 1  or (10_{2415}1_{9}0_{2415}1)  4841
** 6. 10^{4186} + 888 · 10^{2092} + 1  or (10_{2091}8880_{2091}1)  4187
7. 10^{3628} + 88111881818811188 · 10^{1806} + 1 or (10_{1805}881118818188111880_{1805}1) 3629
8. 10^{3504} + 88111818881811188 · 10^{1744} + 1 or (10_{1743}881118188818111880_{1743}1) 3505
* 9. 10^{2992} + 111111111 · 10^{1492} + 1  or (10_{1491}1_{9}0_{1491}1)  2993
* 10. 10^{2810} + 1_{5} · 10^{1403} + 1  or (10_{1402}1_{5}0_{1402}1  2811
* Anti-Yarborough palprime
** Quasi-even-digit palprime

Table 10. B: The TOP TEN Triadic Prime Numbers * [3-way Palprimes] [0, 1, 3, 8 ]

Digits

1. 10^{15640} + 3 · 10^{7820} + 1  or (10_{7819}30_{7819}1)  15641 [H.D: 1999]
2. 10^{11650} + 3 · 10^{5825} + 1  or (10_{5824}30_{5824}1)  11651
3. 10 · 13003 · (10^{9240} − 1)/(10^{5} − 1) + 1  or (13003)_{1848}1  9241
4. 10 · 13388833 · (10^{7560} − 1)/(10^{8} − 1) + 1 or (13388833)_{945}1  7561
5. 10^{6572} + 3 · 10^{3286} + 1 or (10_{3285}30_{3285}1) 6573
6. 10^{5030} + 11101310111 · 10^{2510} + 1  or (10_{2509}111013101110_{2509}1) 5031
7. 10^{4594} + 3 · 10^{2297} + 1  or (10_{2296}30_{2296}1)  4595
8. 10^{3830} + 3 · 10^{1915} + 1  or (10_{1914}30_{1914}1)  3831
9. 10^{3054} + 131 · 10^{1526} + 1  or (10_{1525}1310_{1525}1)  3055
10. 10^{2976} + 3 · 10^{1488} + 1  or (10_{1487}30_{1487}1)  2977
* All discovered by H. Dubner

Table 10. C: The TOP TEN Quasi-Even-Digits Palindromic Prime Numbers

[ Q-E-D Palprimes; 0, 2, 4, 6, 8 ]

Digits E-D%
*** 1. 10^{28572} + 8 · 10^{14286} + 1 or (10_{14285}80_{14285}1) 28573 99.99 [D. Heur: 2001]
2. 10^{15550} + 4260624 · 10^{7772} + 1 or (10_{7771}42606240_{7771}1) 15551 99.99
3. 10^{7058} + 2220222 · 10^{3526} + 1 or (10_{3525}22202220_{3525}1) 7059 99.97
* 4. 10^{5744} + 6_{15} · 10^{2865} + 1  or (10_{2864}6_{15}0_{2864}1)  5745 99.97
** 5. 10^{5250} + 666 · 10^{2624} + 1  or (10_{2623}6660_{2623}1)  5251 99.96
** 6. 10^{4948} + 666 · 10^{2473} + 1  or (10_{2472}6660_{2472}1)  4949 99.96
* 7. 10^{4784} + 6_{15} · 10^{2385} + 1  or (10_{2384}6_{15}0_{2384}1)  4785 99.96
*** 8. 10^{4186} + 888 · 10^{2092} + 1  or (10_{2091}8880_{2091}1)  4187 99.95
* 9. 10^{3322} + 6_{21} · 10^{1651} + 1  or (10_{1650}6_{21}0_{1650}1)  3323 99.94
* 10. 10^{2752} + 6_{15} · 10^{1369} + 1  or (10_{1368}6_{15}0_{1368}1)  2753 99.93
* Generalized beastly palindrome
** Beastly palindrome
*** Tetradic or 4-way prime

Table 10. D: The TOP TEN Zero-free Palindromic Primes

[ 1, 2, 3, 4, 5, 6, 7, 8, 9 ]

Digits

* 1. (9_{6768}29_{6768})  13537 [H.Dubner:1999]
* 2. (9_{5876}29_{5876})  11753
3. (1818535818)_{1008}1 10081
** 4. (159795)_{1680}1  10081
*** 5. (13388833)_{945}1  7561
* 6. (9_{2874}29_{2874})  5749
**** 7. (1676)_{1170}1  4681
* 8. (9_{1918}29_{1918})  3837
**** 9. (1676)_{948}1  3793
10. (11925291)_{450}1  3601
* Near-repdigit prime
** All odd digits
*** Triadic prime
**** Alternate digit prime

Table 10. E: The TOP TEN Near-Repdigit Palprimes

[A_{N}BA_{N}]

Digits

1. 9R(13537) − 7 · 10^{6768} or (9_{6768}29_{6768}) 13537 [H.Dubner: 1999]
2. 9R(11753) − 7 · 10^{5876} or (9_{5876}29_{5876}) 11753
3. 10^{5749} − 7 · 10^{2874} − 1 or (9_{2874}29_{2874}) 5749
4. 10^{3837} − 7 · 10^{1918} − 1 or (9_{1918}29_{1918}) 3837
* 5. 10^{3597} − 10^{1798} − 1  or (9_{1798}89_{1798}) 3597
** 6. 10^{3159} − 8 · 10^{1579} − 1 or (9_{1579}19_{1579}) 3159
** 7. 10^{3017} − 8 · 10^{1508} − 1 or 9_{1508}19_{1508}) 3017
8. 10^{2631} − 7 · 10^{1315} − 1 or (9_{1315}29_{1315}) 2631
* 9. 10^{2493} − 10^{1246} − 1  or (9_{1246}89_{1246}) 2493
*** 10. 10^{2273} − 5 · 10^{1136} − 1 or (9_{1136}49_{1136}) 2273

* All composite digits and curved digits
** All odd digits
*** All composite digits

Table 10. F: The TOP TEN All-Odd Digits Palprimes

[ 1, 3, 5, 7, or 9 ] Digits

1. (159795)_{1680}1 10081 [H.Dubner:1996]
* 2. 9_{1579}19_{1579}  3159
3. (13919193)_{390}1 3121
4. (9_{1508}19_{1508})  3017
**5. (37)_{1441}3  2883
*** 6. (35_{1973}3)  1975
7. (19)_{984}1  1969
8. (37)_{946}3  1893
9. (15)_{895}1  1791
10. (1_{874}91_{874})  1749
* Largest all odd, near-repdigit palprime
** Largest, smoothly undulating prime with prime digits
*** Plateau prime with prime digits

Table 10. G: The Top Beastly Palindromic Prime Numbers*

[(10^{n} + \underline{666}) · 10^{n−2} + 1]

Digits

1. (10^{2626} + 666) · 10^{2624} + 1 or (10_{2623}6660_{2623}1) 5251 [H. Dubner: 1988]
2. (10^{2475} + 666) · 10^{2473} + 1 or (10_{2472}6660_{2472}1) 4949
3. (10^{611} + 666) · 10^{609} + 1 or (10_{608}6660_{608}1)  1221
4. (10^{509} + 666) · 10^{507} + 1 or (10_{506}6660_{506}1)  1017
5. (10^{45} + 666) · 10^{43} + 1  or (10_{42}6660_{42}1)  89
6. (10^{16} + 666) · 10^{14} + 1  or (10_{13}6660_{13}1)  31
7. (10^{3} + 666) · 10 + 1  or (16661)  5

* All discovered by H.Dubner

Table 10. G. 1: The TOP TEN Generalized Beastly Palindromic Primes

[10 … \underline{X666X} … 01]

Digits

1. 10^{9008} + 2166612 · 10^{4501} + 1  or (10_{4500}21666120_{4500}1)  9009 [H.Dubner1990]
2. 10^{5744} + 6_{15} · 10^{2865} + 1  or (10_{2864}6_{15}0_{2864}1)  5745
3. 10^{4784} + 6_{15} · 10^{2385} + 1  or (10_{2384}6_{15}0_{2384}1)  4785
4. 10^{3322} + 6_{21} · 10^{1651} + 1  or (10_{1650}6_{21}0_{1650}1)  3323
5. 10^{2752} + 6_{15} · 10^{1369} + 1  or (10_{1368}6_{15}0_{1368}1)  2753
6. 10^{2692} + 666999666999666 · 10^{1339} + 1 or (10_{1338}6_{3}9_{3}6_{3}9_{3}6_{3}0_{1338}1) 2693
7. 10^{2414} + 10536663501 · 10^{1202} + 1  or (10_{1201}105366635010_{1201}1) 2415
8. 10^{2046} + 6_{15} · 10^{1016} + 1  or (10_{1015}6_{15}0_{1015}1)  2047
9. 10^{1888} + 6660066600666 · 10^{938} + 1  or (10_{937}66600666006660_{937}1) 1889
10. 10^{1812} + 6_{21} · 10^{896} + 1  or (10_{895}6_{21}0_{895}1)  1813

Table 10. H: The TOP TEN Pandigital, Palindromic Prime Numbers* [ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ]

Digits

1. 10^{4954} + 976543282345679 · 10^{2470} + 1  or (10_{2469}9765432823456790_{2469}1)  4955 HD:1987
2. 10^{3826} + 234567989765432 · 10^{1906} + 1  or (10_{1905}2345679897654320_{1905}1)  3827
3. 10^{3802} + 23456789198765432 · 10^{1893} + 1 or (10_{1892}234567891987654320_{1892}1) 3803
4. 10^{3106} + 987654323456789 · 10^{1546} + 1 or (10_{1545}9876543234567890_{1545}1) 3107
5. 10^{2594} + 23456789198765432 · 10^{1289} + 1 or (10_{1288}234567891987654320_{1288}1) 2595
6. 10^{2468} + 23456789198765432 · 10^{1226} + 1 or (10_{1225}234567891987654320_{1225}1) 2469
7. 10^{2220} + 56789123432198765 · 10^{1102} + 1 or (10_{1101}56789123432198765_{1101}1) 2221
8. 10^{1890} + 23456790809765432 · 10^{937} + 1 or (10_{936}234567908097654320_{936}1) 1891
9. 10^{1612} + 23456789298765432 · 10^{798} + 1 or (10_{797}234567892987654320_{797}1) 1613
10. 10^{1522} + 234567989765432 · 10^{754} + 1  or (10_{753}2345679897654320_{753}1)  1523
* All discovered by H. Dubner

Table 10. I: The TOP TEN Alternate-Digit Palprimes

[ odd-even-odd... -odd ]

Digits

1. (1676)_{1170}1 4681 [H. Dubner: 1997]
2. (1676)_{948}1 3793
*3. (12)_{989}1  1979
**4. (1858)_{414}1 1657
**5. (1838)_{410}1 1641
*6. (14)_{815}1  1631
7. (1232)_{402}1 1609
*8. (12)_{798}1  1597
*9. (18)_{739}1  1479
*10. (12)_{699}1  1399
* Smoothly undulating
** Undulating

Table 10. I. 1: The TOP TEN Primes w Alternating Unholey/Holey Digits

[ 1, 2, 3, 5, 7/0, 4, 6, 8, 9 ] Digits

1. (1676)_{1170}1 4681 [H. Dubner: 1997]
2. (1676)_{948}1  3793
*3. (19)_{984}1  1969
**4. (1858)_{414}1  1657
**5. (1838)_{410}1  1641
*6. (14)_{815}1  1631
*7. (18)_{739}1  1479
*8. (16)_{480}1  961
9. (1474)_{231}1  925
**10. (1434)_{205}1  821
* Smoothly Undulating
** Undulating

Table 10. I. 2: The TOP TEN Primes w. Alternating Straight/Curved Digits

[ 1, 4, 7 / 3, 6, 8, 9, 0 ] Digits

1. (1676)_{1170}1 4681 [H. Dubner: 1997]
2. (1676)_{948}1  3793
*3 (37)_{1441}3  2883
*4. (19)_{984}1  1969
*5 (37)_{946}3  1893
*6. (18)_{739}1  1479
*7. (16)_{480}1  961
*8. (37)_{424}3  849
*9. (37)_{157}3  315
*10. (16)_{114}1  229
* Smoothly undulating

Table 10. I. 3: The TOP TEN Primes with

Alternating Prime/Composite Digits [ 2, 3, 5, 7 / 4, 6, 8, 9 ]

Digits

1. (92)_{97}9  195 [H. Dubner: 1991]
2. (78)_{47}7  95
3. (38)_{28}3  57
4. (97)_{22}9  45
5. (97)_{13}9  27
6. (78)_{13}7  27
7. (78)_{10}7  21
8. (38)_{10}3  21
9. (95)_{8}9  17
10. (74)_{8}7  17

Table 10. J: The TOP TEN Palprimes w. Composite Digits

& The TOP TEN Holey Palprimes

[ 4, 6, 8, 9; 0 ]

Digits Holes H/D %
1. 9_{1798}89_{1798}  3597 3598 100.03% [H. Dubner: 1989]
2. 9_{1246}89_{1246}  2493 2494 100.04%
3. 9_{1136}49_{1136}  2273 2273 100.00%
3 A. 9_{874}8089_{874}  1751 1753 100.11%
* 3 B. 9_{593}400049_{593} 1191 1191 100.00%
** 4. 98_{983}9  985 1968 199.80%
5. 9_{378}89_{378}  757 758 100.13%
6. 9_{104}49_{104}  209 209 100.00%
7. 98_{203}9  205 408 199.02%
8. (98)_{80}9  161 241 149.69%
9. (94)_{71}9  143 143 100.00%
10. 98_{113}9  115 228 198.26%

* Holey palprime only
** Highest holes/digits percentage

Table 10. K: The TOP TEN Palprimes w. Curved Digits [ 0, 3, 6, 8, 9 ] Digits

* 1. 9_{1798}89_{1798} 3597 [H. Dubner: 1989]
* 2. 9_{1246}89_{1246} 2493
* 3. 98_{983}9  985
* 4. 9_{378}89_{378}  757
5. 38_{631}3  633
6. 38_{289}3  291
* 7. 98_{203}9  205
8. 3_{94}83_{94}  189
9. 3_{85}83_{85}  171
* 10. (98)_{80}9  161
* All composite digits

Table 10. L: The TOP TEN Unholey Palprimes [ 1, 2, 3, 5, 7 ]

or  Digits
* 1. [(17275727273727273727275727)R_{3120}/R_{26}] · 10 + 1  (17275727273727273727275727)_{120}1  3121 H.D:1992
2. [(1(2)_{17}35553(2)_{17})(R_{3120}/R_{40})] · 10 + 1  (1(2)_{17}35553(2)_{17})_{78}1  3121
**3. 370 · ((100^{1441}) − 1)/99 + 3  (37)_{1441}3  2883
* 4. [(173737573725727257272737573737)R_{2160}/R_{30}] · 10 + 1 (173737573727572727572737573737)_{72}1 2161
* 5. [(173737572727375757372727573737)R_{2160}/R_{30}] · 10 + 1 (173737572727375757372727573737)_{72}1 2161
* 6. [(173727375757572727575757372737)R_{2160}/R_{30}] · 10 + 1 (173727375757572727575757372737)_{72}1 2161
** 7. 120 · R(1978)/R(2) + 1  (12)_{989}1  1979
*** 8. 32 · R(1974) + 1  35_{1973}3  1975
**9. 370 · ((100^{946}) − 1)/99 + 3  (37)_{946}3  1893
** 10. 150 · R(1790)/R(2) + 1  (15)_{895}1  1791
* Undulating prime
** Smoothly undulating prime
*** Largest prime with prime digits

Table 10. M: The TOP TEN Invertible Palprime Pairs

[ 0, 1, 6, 8, 9 ]  Digits
1A. (10_{1137}88088108180898081801880880_{1137}1)  2301 H. Dubner: 2000
1B. (10_{1137}88088108180868081801880880_{1137}1)  2301
2A. (10_{1038}801111181089801811111080_{1038}1)  2101
2B. (10_{1038}801111181086801811111080_{1038}1)  2101
3A. (10_{748}800880081819181800880080_{748}1)  1521
3B. (10_{748}800880081816181800880080_{748}1)  1521
4A. (10_{742}18180101800898008101081810_{742}1)  1511
4B. (10_{742}18180101800868008101081810_{742}1)  1511
5A. (10_{736}8101880180118981108108810180_{736}1)  1501
5B. (10_{736}8101880180118681108108810180_{736}1)  1501
6A. (10_{735}100088010081819181800108800010_{735}1) 1501
6B. (10_{735}100088010081816181800108800010_{735}1) 1501
7A. (10_{685}808008808881009001888088008080_{685}1) 1401
7B. (10_{685}808008808881006001888088008080_{685}1) 1401
8A. (10_{685}181188011110189810111108811810_{685}1) 1401
8B. (10_{685}181188011110186810111108811810_{685}1) 1401
9A. (10_{685}111110888080189810808880111110_{685}1) 1401
9B. (10_{685}111110888080186810808880111110_{685}1) 1401
10A. (10_{685}100080108088089808808010800010_{685}1) 1401
10B. (10_{685}100080108088086808808010800010_{685}1) 1401

Table 10. N: The TOP TEN Smoothly Undulating Palprimes

[AB_{N}A] Digits

1.  (37)_{1441}3  2883 [L.C. Noll: 1997]
2.  (12)_{989}1  1979
3.  (19)_{984}1  1969
4.  (37)_{946}3  1893
5.  (15)_{895}1  1791
6.  (14)_{815}1  1631
7.  (12)_{798}1  1597
8.  (18)_{739}1  1479
9.  (12)_{699}1  1399
10.  (16)_{480}1  961

Table 10. O: The TOP TEN Palprimes w. Prime Digits

[ 2, 3, 5, 7 ]

Digits

* 1. (37)_{1441}3 2883 [L.C. Noll:1997]
** 2. 35_{1973}3  1975
* 3. (37)_{946}3  1893
*** 4. 32_{893}3  895
* 5. (37)_{424}3  849
***** 6. (2_{P} + 1)  727
** 7. 35_{725}3  727
*** 8. 72_{723}7  725
**** 9. 7_{253}57_{253}  507
** 10. 35_{461}3  463
*  Smoothly undulating prime
**  Plateau prime
***  Depression prime
****  Near-repdigit prime
*****  Undulating prime: to prove a prime (p) to be of the Sophie Germain

type, (2p + 1) must also be found prime (H.Dubner: 1999)

(2p + 1)=

7272727232323272323232723272327272327232727272327232323272723232323232 3232323232323232323232323232323232323232323232323232323232323232323232 3232323232323232323232323232323232323232323232323232323232323232323232 3232323232323232323232323232323232323232323232323232323232323232323232 3232323232323232323232323232323232323232323232323232323232323232323232 3232323232323232323232323232323232323232323232323232323232323232323232 3232323232323232323232323232323232323232323232323232323232323232323232 3232323232323232323232323232323232323232323232323232323232323232323232 3232323232323232323232323232323232323232323232323232323232323232323232 3232323232323232323232323232323232323272723232327232727272327232727232 723272323232723232327272727 Table 10. P: The TOP TEN Palindromic Quasi-Repdigit Prime Numbers

(AB_{n}A)

Repdigits  Repd. %
1. 35_{1973}3  1973  99.90  [C.Rivera:1997]
2. 13_{1469}1  1469  99.86
3. 17_{1001}1  1001  99.80
4. 98_{983}9  983  99.80
5. 32_{893}3  893  99.78
6. 18_{883}1  883  99.77
7. 19_{729}1  729  99.73
8. 35_{725}3  725  99.72
9. 72_{723}7  723  99.72
10. 38_{631}3  631  99.68

Table 10. P. 1: The TOP TEN P^{latea}u Prime Numbers

(AB_{n}A), A < B
Repdigits  Repd. %
1. 35_{1973}3  1973  99.90  [C.Rivera:1997]
2. 13_{1469}1  1469  99.86
3. 17_{1001}1  1001  99.80
4. 18_{883}1  883  99.77
5. 19_{729}1  729  99.73
6. 35_{725}3  725  99.72
7. 38_{631}3  631  99.68
8. 78_{565}7  565  99.65
9. 15_{561}1  561  99.64
10. 34_{491}3  491  99.59

Table 10. P. 2: The TOP TEN D_{epressio}n Prime Numbers

[ (AB_{n}A), A > B ]
Repdigits  Repd. %
1. 98_{983}9  983  99.80  [R. Carr, etal: 2000]
2. 32_{893}3  893  99.78
3. 72_{723}7  723  99.72
4. 31_{599}3  599  99.67
5. 76_{573}7  573  99.65
6. 74_{483}7  483  99.59
7. 76_{453}7  453  99.56
8. 75_{421}7  421  99.53
9. 75_{349}7  349  99.43
10. 31_{341}3  341  99.42

Table 10. Q: The TOP TEN Palprimes w. Straight Digits

[ 1, 4, 7 ] Digits

1. (14)_{815}1  1631 [C. Rivera:1997]
* 2. R_{1031}  1031
3. 17_{1001}1  1003
4. (1474)_{231}1  925
5. (14)_{291}1  583
6. 74_{483}7  485
7. (14)_{239}1  479
8. 17_{365}1  367
9. R_{317}  317
10. (14)_{138}1  277
* Largest known repunit prime: (10^{1031} − 1)/9

Table 10. R: The Five Repunit Primes Known * [ R_{n} = (10^{n} − 1)/9 = 1_{n} ] Digits

1. R_{1031} = (10^{1031} − 1)/9 or 1_{1031} 1031 [ H.C. Williams & H.Dubner:1985]
2. R_{317} = (10^{317} − 1)/9 or 1_{317}  317
3. R_{23} = (10^{23} − 1)/9  or 1_{23}  23
4. R_{19} = (10^{19} − 1)/9  or 1_{19}  19
5. R_{2} = (10^{2} − 1)/9  or 1_{2}  2
* All other repunits up to R(49080) are composite

Table 10. S: The TOP TEN Known Palindromic Primes in Arithmetic Progression 27 Digits

1. 742 950 290 879 090 978 092 059 247 [H.Dubner, T.Forbes, M.Toplic, etal: 1999]
2. 742 950 290 878 080 878 092 059 247
3. 742 950 290 877 070 778 092 059 247
4. 742 950 290 876 060 678 092 059 247
5. 742 950 290 875 050 578 092 059 247
6. 742 950 290 874 040 478 092 059 247
7. 742 950 290 873 030 378 092 059 247
8. 742 950 290 872 020 278 092 059 247
9. 742 950 290 871 010 178 092 059 247
10. 742 950 290 870 000 078 092 059 247

Common difference = 1 010 100 000 000 000 Table 11: The TOP TEN Primes with Square Digits

[ 0, 1, 4, 9 ]

Digits

* 1. 10_{15397}11101110_{15397}1 30803 [H.Dubner:1999]
** 2. 19_{19233}  19234
** 3. 49_{16131}  16132
** 4. 49_{15796}  15797
** 5. 19_{15749}  15750
6. 4_{10200}0_{2893}1  13094
7. 1_{10080}0_{2135}1  12216
8. 9_{10080}0_{850}1  10931
9. 190_{10146}1  10149
10. 4_{9240}0_{150}1  9391
* Tetradic or 4-way prime
** Near repdigit string prime

Table 12: The TOP TEN Prime Numbers w. Cube Digits **

[ 0, 1, 8 ]

Digits

* 1. (10_{15397}11101110_{15397}1)  30803 [H. Dubner:1999]
* 2. (10_{14285}80_{14285}1  28573
3. 8_{12600}0_{3704}1  16305
4. 1_{10080}0_{2135}1  12216
5. 8_{10080}0_{1002}1  11083
* 6. (110101)_{1680}1  10081
7. 1_{6300}0_{2137}1  8438
* 8. (10_{3444}881118818188111880_{3444}1) 6907
9. 1_{2700}0_{3155}1  5856
10. 1_{2502}0_{2611}1  5114
* Tetradic or 4-way prime
** All discovered by H. Dubner

Table 13: The TOP TEN Anti-Yarborough Primes with 1’ s and 0’ s Digits **

Digits

* 1. (10_{15397}11101110_{15397}1) 30803 [H. Dubner:1999]
2. 1_{10080}0_{2135}1  12216
* 3. (110101)_{1680}1  10081
4. 1_{6300}0_{2137}1  8438
5. 1_{2700}0_{3155}1  5856
6. 1_{2502}0_{2611}1  5114
7. 1_{2502}0_{2501}1  5004
* 8. (10_{2415}1_{9}0_{2415}1)  4841
9. 1_{3120}0_{210}1  3331
10. 1_{2062}0_{1051}1  3114

* Tetradic or 4-way prime
** All discovered by H. Dubner

Table 14: The TOP TEN Yarborough Prime Numbers and The TOP TEN Zero-Free Primes [ 2, 3, 4, 5, 6, 7, 8, 9; 1 ]

Digits

*1. 10^{50103} − 4 · 10^{50097} − 1  or 9_{5}(5)9_{50097}  50103 [P. Carmody:2000]
**2. 9 · 10^{48051} − 1  or (8)9_{48051}  48052
**3. 9 · 10^{41475} − 1  or (8)9_{14175}  41476
*4. 10^{38500} − 10^{18168} − 1  or 9_{20332}(8)9_{18168} 38501
**5. 3 · 10^{33058} − 1 or (2)9_{33058} 33059
*6. 10^{30007} − 10^{22717} − 1  or 9_{7290}(8)9_{22717} 30008
*7. 10^{30006} − 10^{21425} − 1  or 9_{8581}(8)9_{21425} 30007
*8. 10^{30005} − 10^{23906} − 1  or 9_{6099}(8)9_{23906} 30006
*9. 10^{30004} − 10^{16623} − 1  or 9_{13380}(8)9_{16623} 30004
*9. 10^{30004} − 10^{17794} − 1  or 9_{12209}(8)9_{17794} 30004
*9. 10^{30003} − 10^{16681} − 1  or 9_{13322}(8)9_{16681} 30004
*9. 10^{30003} − 10^{17640} − 1  or 9_{12363}(8)9_{17604} 30004
* Near-repdigit prime
** Near-repdigit string prime
() The “Odd” digit of that prime

Table 15: The TOP TEN Prime Twins [K · 2^{N} ± 1] Digits

1.  1807318575 · 2^{98305} ± 1  29603 [D. Underbakke, P. Carmody: 2001]
2.  665551035 · 2^{80025} ± 1  24099
3.  1693965 · 2^{66443} ± 1  20008
4.  83475759 · 2^{64955} ± 1  19562
5. 4648619711505 · 2^{60000} ± 1  18075
6. 2409110779845 · 2^{60000} ± 1  18075
7. 2230907354445 · 2^{48000} ± 1  14462
8. 871892617365 · 2^{48000} ± 1  14462
9.  361700055 · 2^{39020} ± 1  11755
10.  835335 · 2^{39014} ± 1  11751

Table 15. A: The TOP TEN Prime Triplets

[k − tuplets, k = 3]

Digits

1. p-1, p+1, p+5 1852468459 · 4999#/35 2141 [H. Rosenthal & P. Jobling:2001]
2. p-1, p+1, p+5 1042334284 · 4999#/35 2141
3. p-1, p+1, p+5 177299114 · 4999#/35 2140
4. p-1, p-1, p+5 1279378536 · 4993#/35 2137
5. p-1, p+1, p+5 508157676 · 4993#/35 2137
6. p-1, p+1, p+5 122194876 · 4983#/35 2136
7. p-1, p+1, p+5 1855266543 · 4987#/35 2134
8. p-1, p+1, p+5 167761138 · 4987#/35 2134
9. p-5, p-1, p+1 871453243 · 4987#/35 2134
10. p-1, p+1, p+5 388838923 · 4987#/35 2133

p# = 2 · 3 · 5 · 7 · 11 … p Table 15. B: The Top K-tuplets of Primes [k > 3] [Prime Quadruplets, Quintuplets, Sextuplets, etc.]

Digits

1. 4-tuplets:
p-7, p-5, p-1, p+1 P = 8954571083387140525(2^{3423} − 2^{1141}) − 6 · 2^{1141} 1050 [T. Forbes: 1999]
2. p-7, p-5, p-1, p+1  P = 24947432928741915235(2^{3363} − 21121) − 6 · 2^{1121} 1032
3. p-7, p-5, p-1, p+1  P = 17293378403589618790(2^{3363} − 2^{1121}) − 6 · 2^{1121}  1032
4. p-7, p-5, p-1, p+1  P = 11984747204231082960(2^{3363} − 2^{1121}) − 6 · 2^{1121}  1032
5. p-7, p-5, p-1, p+1  P = 3510160221387831655(2^{3363} − 2^{1121}) − 6 · 2^{1121}  1031
6. p-7, p-5, p-1, p+1  P = 331426625784936325(2^{3363} − 2^{1121}) − 6 · 2^{1121}  1030
7. p-7, p-5, p-1, p+1  P = 76912895956636885(2^{3279} − 2^{1093}) − 6 · 2^{1093}  1004

5-tuplets:

8. p, p+4, p+6, p+10, p+12 P = 3242281037 · 900# + 1867 384 [M. Bell: 2000]

6-tuplets:

* 9. p, p+4, p+6, p+10, p+12, p+16 p = 97953153175 · 670# + 16057 290 [M.Bell, etal: 2001]

7-tuplets:

* 10. p, p+2, p+8, p+12, p+14, p+18, p+20 P = 60922342070 · 350# + 5639  152 [M.Bell: 2001]
* p^{#} = 2 · 3 · 5 · 7 · 11 … p

Table 16. A: The TOP TEN Near Repdigit Prime Numbers AB_{n} [Streak or String Primes: ABBBBBBBB…B]

Repdigits

*1. 9 · 10^{48051} − 1  or 89_{48051} 48051 [H. Dubner & Y. Gallot: 2000]
* 2. 9 · 10^{41475} − 1  or 89_{41475} 41475
3. 3 · 10^{33058} − 1  or 29_{33058} 33058
4. 3 · 10^{26044} − 1  or 29_{26044} 26044
** 5. 2 · 10^{19233} − 1  or 19_{19233} 19233
** 6. 6 · 10^{18668} − 1  or 59_{18668} 18668
* 7. 5 · 10^{16131} − 1  or 49_{16131} 16131
* 8. 5 · 10^{15796} − 1  or 49_{15796} 15796
** 9. 2 · 10^{15749} − 1  or 19_{15749} 15749
** 10. 8 · 10^{11336} − 1  or 79_{11336} 11336
* All composite digits
** All odd digits

Table 16. B: The TOP TEN Near-Repdigit Prime Numbers A_{n−k−1}BA_{k}

Digits

1. 10^{50103} − 4 · 10^{50097} − 1 or 9_{5}59_{50097} 50103 [P. Carmody: 2000]
2. 10^{38500} − 10^{18168} − 1  or 9_{20332}89_{18168} 38501
3. 10^{30007} − 10^{22717} − 1  or 9_{7290}89_{22717} 30008
4. 10^{30006} − 10^{21425} − 1  or 9_{8581}89_{21225} 30007
5. 10^{30005} − 10^{23906} − 1  or 9_{6099}89_{23906} 30006
6. 10^{30004} − 10^{16623} − 1  or 9_{13380}89_{16623} 30004
7. 10^{30004} − 10^{17794} − 1  or 9_{12209}89_{17794} 30004
8. 10^{30003} − 10^{16681} − 1  or 9_{13322}89_{16681} 30004
9. 10^{30003} − 10^{17640} − 1  or 9_{12363}89_{17640} 30004
10. 10^{30002} − 10^{21020} − 1  or 9_{8982}89_{21020} 30003

Table 16. C: The TOP TEN Near Repdigit Prime Numbers A_{n}B

[A_{N}B]

[Streak or string primes: …AAAAAAAAAB]

Repdigits Repd %
1. 30 · R(1917) + 1  or 3_{1917}1  1917  99.95 [C. Rivera: 1997]
2. 30 · R(1731) + 1  or 3_{1731}1  1731  99.94
3. 90 · R(1656) + 1  or 9_{1656}1  1656  99.94
4. 60 · R(1599) + 1  or 6_{1599}1  1599  99.94
5. 20 · R(1493) + 1  or 2_{1493}1  1493  99.93
6. 80 · R(1421) + 7  or 8_{1421}7  1421  99.93
7. 80 · R(1418) + 1  or 8_{1418}1  1418  99.93
8. 70 · R(1368) + 1  or 7_{1368}1  1368  99.93
9. 60 · R(1363) + 7  or 6_{1363}7  1363  99.93
10. 20 · R(1216) + 3  or 2_{1216}3  1216  99.92

Table 16. D: The TOP TEN Near-Repunit Prime Numbers

[1_{n−k−1}01_{k} ]

Digits

1. R(179) · 10^{1485} + R(1484)  or 1_{179}01_{1484} 1664 [C. Rivera: 1997]
2. R(396) · 10^{686} + R(685)  or 1_{396}01_{685}  1082
3. R(359) · 10^{721} + R(720)  or 1_{359}01_{720}  1080
4. R(370) · 10^{632} + R(631)  or 1_{370}01_{631}  1002
5. R(29) · 10^{973} + R(972)  or 1_{29}01_{972}  1002
6. R(741) · 10^{260} + R(261)  or 1_{740}21_{260}  1001
7. R(534) − 10^{178}  or 1_{355}01_{178}  534
8. R(381) − 10^{127}  or 1_{253}01_{127}  381
9. R(332) − 10^{111}  or 1_{220}01_{111}  332
10. R(282) − 10^{188}  or 1_{93}01_{188}  282

Table 17: The TOP TEN Quasi-Repdigit Prime Numbers

[k · 10^{n} + 1]
Repdigits  Repd. %
* 1. 3 · 10^{27720} + 1  or 30_{27719}1  27719  99.99  [J. Liddle & Y. Gallot: 2000]
* 2. 3 · 10^{10453} + 1  or 30_{10452}1  10452  99.98
** 3. 6 · 10^{4426} + 1  or 60_{4425}1  4425  99.95
** 4. 6 · 10^{2629} + 1  or 60_{2628}1  2628  99.92
* 5. 3 · 10^{2620} + 1  or 30_{2619}1  2619  99.92
** 6. 6 · 10^{2236} + 1  or 60_{2235}1  2235  99.91
* 7. 7 · 10^{2196} + 1  or 70_{2195}1  2195  99.91
*** 8. 32 · R(1974) + 1  or 35_{1973}3  1973  99.90
* 9. 3 · 10^{1900} + 1  or 30_{1899}1  1899  99.89
**** 10. 12 · R(1470) − 1  or 13_{1469}1  1469  99.86

* Quasi-even-digits prime
** Almost-all-even-digits prime
*** Plateau palindrome with all prime digits
**** Plateau palindrome

Table 18: The TOP TEN Factorial Prime Numbers

[N! ± 1 = 1 × 2 × 3 × 4 … N ± 1]
N! ± 1  Digits
1. 6917! − 1  23560 [C. Caldwell & Y.Gallot: 1998]
2. 6380! + 1  21507
3. 3610! − 1  11277
4. 3507! − 1  10912
5. 1963! − 1  5614
6. 1477! + 1  4042
7. 974! − 1  2490
8. 872! + 1  2188
9. 546! − 1  1260
10. 469! − 1  1051

Table 18. A: The TOP TEN Factorial -Plus-One Primes

[N! + 1 = 1 × 2 × 3 × … N + 1]
N! + 1  Digits
1. 6380! + 1  21507 [C. Caldwell & Y. Gallot:1998]
2. 1477! + 1  4042
3. 872! + 1  2188
4. 427! + 1  940
5. 399! + 1  867
6. 340! + 1  715
7. 320! + 1  665
8. 154! + 1  272
9. 116! + 1  191
10. 77! + 1  114

[ and the rest ]

N! + 1 Digits  N! + 1 Digits
11. 73! + 1  106  15. 11! + 1  8
12. 41! + 1  50  16. 3! + 1  1
13. 37! + 1  44  17. 2! + 1  1
14. 27! + 1  29  18. 1! + 1  1

Table 18. B: The TOP TEN Factorial -Minus-One Primes

[N! − 1 = 1 × 2 × 3 × … N − 1]
N! − 1  Digits
1. 6917! − 1 23560 [C. Caldwell & Y. Gallot: 1998]
2. 3610! − 1 11277
3. 3507! − 1 10912
4. 1963! − 1 5614
5. 974! − 1  2490
6. 546! − 1  1260
7. 469! − 1  1051
8. 379! − 1  815
9. 324! − 1  675
10. 166! − 1  298

[and the rest]

N! − 1 Digits  N! − 1 Digits
11. 94! − 1  147  17. 12! − 1  9
12. 38! − 1  45  18. 7! − 1  4
13. 33! − 1  37  19. 6! − 1  3
14. 32! − 1  36  20. 4! − 1  2
15. 30! − 1  33  21. 3! − 1  1
16. 14! − 1  11

Table 19: The TOP TEN Primes of Alternating Sums of Factorials

[A_{n} = n! − (n − 1)! + (n − 2)! − + ⋯ − (−1)^{n}1!]

Digits

1. 160! − 159! + 158! − ... − 3! + 2! − 1!  285 [W.Keller:
2. 105! − 104! + 103! − ... + 3! − 2! + 1!  169
3. 61! − 60! + 59! − ... + 3! − 2! + 1!  84
4. 59! − 58! + 57! − ... + 3! − 2! + 1!  81
5. 41! − 40! + 39! − ... + 3! − 2! + 1!  50
6. 19! − 18! + 17! − ... + 3! − 2! + 1!  18
7. 15! − 14! + 13! − ... + 3! − 2! + 1!  13
8. 10! − 9! + 8! − ... − 3! + 2! − 1!  7
9. 8! − 7! + 6! − 5! + 4! − 3! + 2! − 1!  5
10. 7! − 6! + 5! − 4! + 3! − 2! + 1!  4

[and the rest]

11. 6! − 5! + 4! − 3! + 2! − 1!  3
12. 5! − 4! + 3! − 2! + 1!  3
13. 4! − 3! + 2! − 1!  2
14. 3! − 2! + 1!  1

Table 20: The TOP TEN Primorial Prime Numbers

[P^{#} ± 1 = 2 × 3 × 5 × 7 × … P ± 1]

Digits

1. 145823^{#} + 1 63142 [Anderson & Robinson: 2000]
2. 42209^{#} + 1 18241
3. 24029^{#} + 1 10387
4. 23801^{#} + 1 10273
5. 18523^{#} + 1 8002
6. 15877^{#} − 1 6845
7. 13649^{#} + 1 5862
8. 13033^{#} − 1 5610
9. 11549^{#} + 1 4951
10.  6569^{#} − 1 2811

Table 20. A: The TOP TEN Primorial-Plus-One Primes

[P^{#} + 1 = 2 × 3 × 5 × … P + 1]

Digits

1. 145823^{#} + 1 63142 [Anderson & Robinson: 2000]
2. 42209^{#} + 1 18241
3. 24029^{#} + 1 10387
4. 23801^{#} + 1 10273
5. 18523^{#} + 1 8002
6. 13649^{#} + 1 5862
7. 11549^{#} + 1 4951
8.  4787^{#} + 1 2038
9.  4547^{#} + 1 1939
10.  3229^{#} + 1 1368

[ and the rest ]

11.  2657^{#} + 1 1115
12.  1021^{#} + 1  428
13.  1019^{#} + 1  425
14.  379^{#} + 1  154
15.  31^{#} + 1  12
16.  11^{#} + 1  4
17.  7^{#} + 1  3
18.  5^{#} + 1  2
19.  3^{#} + 1  1
20.  2^{#} + 1  1

Table 20. B: The TOP TEN Primorial-Minus-One Primes

[P^{#} − 1 = 2x3x5x7x … P − 1]

Digits

1. 15877^{#} − 1 6845 [C. Caldwell & H. Dubner: 1992]
2. 13033^{#} − 1 5610
3. 6569^{#} − 1 2811
4. 4583^{#} − 1 1953
5. 4297^{#} − 1 1844
6. 4093^{#} − 1 1750
7. 2377^{#} − 1 1007
8. 2053^{#} − 1  866
9. 1873^{#} − 1  790
10.  991^{#} − 1  413

[and the rest]

11.  337^{#} − 1  136
12.  317^{#} − 1  131
13.  89^{#} − 1  35
14.  41^{#} − 1  15
15.  13^{#} − 1  5
16.  11^{#} − 1  4
17.  5^{#} − 1  2
18.  3^{#} − 1  1

Table 21: The TOP TEN Multifactorial Prime Numbers [N! _{k} ± 1] Digits

1. 34706!!! − 1  47505 [S. Harvey: 2000]
2. 34626!!! − 1  47384
3. 32659!!! + 1  44416
4. 69114! _{7} − 1  43519
5. 28565!!! + 1  38295
6. 61467! _{7} − 1  38238
7. 54481! _{7} − 1  33485
8. 24753!!! + 1  32671
9. 23109!!! − 1  30272
10. 41990! _{6} − 1  29318

Table 21. A: The TOP TEN Double Factorial Prime Numbers

[N !! ± 1 = N · (N − 2) (N − 4) (N − 6) … (2) ± 1]

Digits

1. 9682!! − 1 17196 [B. deWater: 1999]
2. 8670!! − 1 15191
3. 6404!! − 1 10800
4. 3476!! − 1 5402
5. 2328!! − 1 3416
6. 888!! − 1 1118
7. 842!! − 1 1051
8. 728!! − 1  886
9. 518!! + 1  593
10. 214!! − 1  205

Table 21. B: The TOP TEN Triple Factorial Prime Numbers

[N !!! ± 1 = N · (N − 3) (N − 6) (N − 9) ⋯ ± 1]

Digits

1. 34706!!! − 1 47505 [S. Harvey: 2000]
2. 34626!!! − 1 47384
3. 32659!!! + 1 44416
4. 28565!!! + 1 38295
5. 24753!!! + 1 32671
6. 23109!!! − 1 30272
7. 22326!!! + 1 29135
8. 21725!!! + 1 28265
9. 18037!!! + 1 22981
10. 16681!!! + 1 21065

Table 21. C: The TOP TEN Quadruple Factorial Prime Numbers

[N !!!! ± 1 = N · (N − 4) (N − 8) (N − 12) … ± 1]

Digits

1. 27780!!!! − 1  27848 [S. Harvey: 2000]
2. 25938!!!! − 1  25809
3. 22726!!!! − 1  22287
4. 19978!!!! − 1  19313
5. 14614!!!! + 1  13632
6. 12778!!!! − 1  11733
7. 6586!!!! + 1  5575
8. 5920!!!! − 1  4943
9. 5680!!!! + 1  4717
10. 5612!!!! + 1  4653

Table 21. D: The TOP TEN Quintuple Factorial Prime Numbers

[N !!!!! ± 1 = N · (N − 5) (N − 10) (N − 15) ⋯ ± 1]

Digits

1. 22753!!!!! − 1  17854 [B. deWater: 2000]
2. 21092!!!!! − 1  16412
3. 12415!!!!! − 1  9090
4. 12144!!!!! − 1  8868
5. 11915!!!!! + 1  8681
6. 10448!!!!! − 1  7493
7. 10232!!!!! + 1  7320
8. 9992!!!!! − 1  7128
9. 9382!!!!! + 1  6641
10. 9202!!!!! + 1  6498

Table 21. E: The TOP TEN Sextuple Factorial Prime Numbers

[N! _{6} ± 1 = N(N − 6) (N − 12) (N − 18) ⋯ ± 1]

Digits

1. 41990! _{6} − 1  29318 [R. Ballinger: 2000]
2. 38618! _{6} − 1  26730
3. 29882! _{6} − 1  20129
4. 25848! _{6} − 1  17141
5. 25336! _{6} − 1  16764
6. 21906! _{6} − 1  14264
7. 21432! _{6} − 1  13922
8. 12798! _{6} + 1  7837
9. 12760! _{6} + 1  7811
10. 11250! _{6} + 1  6784

Table 21. F: The TOP TEN Septuple Factorial Prime Numbers

[N! _{7} ± 1 = N(N − 7) (N − 14) (N − 21) ⋯ ± 1]

Digits

1. 69144! _{7} − 1  43519 [R. Dohmen: 2000]
2. 61467! _{7} − 1  38238
3. 54481! _{7} − 1  33485
4. 45811! _{7} − 1  27664
5. 43328! _{7} − 1  26015
6. 40707! _{7} − 1  24284
7. 31386! _{7} − 1  18218
8. 27430! _{7} − 1  15693
9. 26598! _{7} − 1  15166
10. 24014! _{7} − 1  13540

Table 22: The TOP TEN Primes with Composite Digits and The TOP TEN Holey Primes [ 4, 6, 8, 9; 0 ]

Digits Holes
1. 89_{48051}  48052  48053 [H. Dubner & Y. Gallot: 2000]
2. 89_{41475}  41476  41477
3. 9_{20332}89_{18168}  38051  38052
4. 9_{7290}89_{22717}  30008  30009
5. 9_{8581}89_{21425}  30007  30008
6. 9_{6099}89_{23906}  30006  30007
7. 9_{13380}89_{16623}  30004  30005
7. 9_{12209}89_{17794}  30004  30005
7. 9_{13322}89_{16681}  30004  30005
7. 9_{12363}89_{17640}  30004  30005

Table 23: The TOP TEN Undulating Prime Numbers

or  Digits
*1. 10 · 1704060407 · R(12600)/R(10) + 1  (1704060407)_{1260}1  12601 [ HD: 1997 ]
*2. 10 · 1506484605 · R_{9240}/R_{10} + 1  (1506484605)_{924}1  9241
*3. [(17275727273727273727275727) · R_{3120}/R_{26}] · 10 + 1  (17275727273727273727275727)_{120}1  3121
**4. (72323252323272325252)(10^{3120} − 1)/(10^{20} − 1) + 1  (72323252323272325252)_{156} + 1  3120
**5. (72323232723232525252)(10^{3120} − 1)/(10^{20} − 1) + 1  (72323232723232525252)_{156} + 1  3120
***6. 370 · ((100^{1441}) − 1)/99 + 3  (37)_{1441}3  2883
*7. [(173737573727572727572737573737)R_{2160}/R30] · 10 + 1 (173737573727572727572737573737)_{72}1 2161
*8. [(173737572727375757372727573737)R_{2160}/R_{30}] · 10 + 1 (173737572727375757372727573737)_{72}1 2161
*9. [(173727375757572727575757372737)R_{2160}/R_{30}] · 10 + 1 (173727375757572727575757372737)_{72}1 2161
**10. (723232523232327272)(10^{2160} − 1)/(10^{18} − 1) + 1  (723232523232327272)_{120} + 1  2160
**10. (523232525252327272)(10^{2160} − 1)/(10^{18} − 1) + 1  (523232525252327272)_{120} + 1  2160

* Palindrome
** Anti-palindrome with prime digits
*** Smoothly undulating palindrome with prime digits

Table 24: The TOP TEN Primes with Curved Digits [ 0, 3, 6, 8, 9 ] Digits

1. 89_{48051}  48052 [H. Dubner & Y. Gallot: 2000]
2. 89_{41475}  41476
3. 9_{20332}89_{18168}  38501
4. 9_{7290}89_{22717}  30008
5. 9_{8581}89_{21425}  30007
6. 9_{6099}89_{23906}  30006
7. 9_{13380}89_{16623}  30004
7. 9_{12209}89_{17794}  30004
7. 9_{13322}89_{16681}  30004
7. 9_{12363}89_{17640}  30004

Table 25: The TOP TEN Sophie Germain Primes

[P = k · 2^{n} − 1 ; p = k# ± 1]

Digits

1. 109433307 · 2^{66452} − 1  20013 [Underbakke, Jobling, Gallot: 2001]
2. 984798015 · 2^{66444} − 1  20011
3. 3714089895285 · 2^{60000} − 1 18075
4. 18131 · 22817# − 1  9853
5. 18458709 · 2^{32611} − 1  9825
6. 415365 · 2^{30052} − 1  9053
7. 1051054917 · 2^{25000} − 1  7535
8. 885817959 · 2^{24711} − 1  7448
9. 1392082887 · 2^{24680} − 1  7439
10. 14516877 · 2^{24176} − 1  7285

p# = 2 · 3 · 5 · 7 · 11 … p Table 26: The TOP TEN Anti-Palindromic Prime Numbers [D · R_{n} · 10^{n} ± 1 & k · R_{n}/R_{a} + 1]

or  Digits
**1. 2 · R_{3038} · 10^{3038} + 1  2_{3038}0_{3037}1  6076 H. D: 1990
**2. 6 · R_{2476} · 10^{2476} + 1  6_{2476}0_{2475}1  4952
3. 5 · R_{2093} · 10^{2093} + 1  5_{2093}0_{2092}1  4186
*4. (72323252323272325252)(10^{3120} − 1)/(10^{20} − 1) + 1 (72323252323272325252)_{156} + 1 3120
*5. (72323232723232525252)(10^{3120} − 1)/(10^{20} − 1) + 1 (72323232723232525252)_{156} + 1 3120
*6. (723232523232327272)(10^{2160} − 1)/(10^{18} − 1) + 1  (723232523232327272)_{120} + 1  2160
*7. (523232525252327272)(10^{2160} − 1)/(10^{18} − 1) + 1  (523232525252327272)_{120} + 1  2160
*8. (523232325232525252)(10^{2160} − 1)/(10^{18} − 1) + 1  (523232325232525252)_{120} + 1  2160
9. 4 · R(975) · 10^{975} − 1  4_{974}39_{975}  1950
10. 3 · R(835) · 10^{835} + 1  3_{835}0_{834}1  1670
* Undulating prime digits
** Almost all even digits

Table 27: The TOP TEN Generalized Repunit Primes *

[ (b^{n} − 1) / (b − 1) ; b ≠ 2 or 10 ]

Digits

1. (7568^{3361} − 1)/7567  13034 [A. Steward: 2001]
2. (8854^{2521} − 1)/8853  9947
3. (7147^{2161} − 1)/7146  8325
4. (5701^{2161} − 1)/5700  8113
5. (4411^{2161} − 1)/4410  7873
6. (3709^{2161} − 1)/3708  7710
7. (6878^{1801} − 1)/6877  6908
8. (675065^{1153} − 1)/675064  6716
9. (402^{2521} − 1)/401  6563
10. (5507^{1621} − 1)/5506  6061

* Excluding Mersenne and repunit primes Table 28: The TOP TEN Strobogrammatic Primes (nonpalindromic)

[ 0, 1, 6, 8, 9 ] Digits

1. 10_{2709}6669990_{2709}1  5426 [H. Dubner: 1996]
2. 10_{2301}6890_{2301}1  4607
3. 6_{1527}19_{1527}  3055
4. 10_{1313}9996660_{1313}1  2634
5. 10_{1295}9996660_{1295}1  2598
6. 10_{899}66689990_{899}1  1807
7. 10_{801}99906660_{801}1  1611
8. 10_{773}6890_{773}1  1551
9. 10_{754}66609990_{754}1  1517
10. 6_{685}19_{685}  1371

Table 29: The TOP TEN Beastly Primes (nonpalindromic) *

[\underline{666} · 10^{n} + 1 = \underline{666}0000 … 0001]

or  Digits
1. 666 · 10^{14020} + 1 6660_{14019}1 14023 [H. Dubner: 2000]
2. 666 · 10^{9198} + 1 6660_{9197}1  9201
3. 666 · 10^{4741} + 1 6660_{4740}1  4744
4. 666 · 10^{3076} + 1 6660_{3075}1  3079
5. 666 · 10^{2928} + 1 6660_{2927}1  2931
6. 666 · 10^{1592} + 1 6660_{1591}1  1595
7. 666 · 10^{718} + 1 6660_{717}1  721
8. 666 · 10^{619} + 1 6660_{618}1  622
9. 666 · 10^{580} + 1 6660_{579}1  583
10. 666 · 10^{373} + 1 6660_{372}1  376

[and the rest]

11. 666 · 10^{48} + 1  6660_{47}1  51
12. 666 · 10^{30} + 1  6660_{29}1  33
13. 666 · 10^{12} + 1  6660_{11}1  15
14. 666 · 10^{10} + 1  6660_{9}1  13
15. 666 · 10^{2} + 1  66601  5
16. 666 · 10 + 1  6661  4

* All discovered by H. Dubner Table 30: The TOP TEN Sub _{script} Prime Numbers *

Digits

1. 1_{1000}2_{1000}3_{1000}4_{1000}5_{1000}6_{1000}7_{1000}8_{1000}9_{1000}0_{6645}1 15646 [H. Dubner: 2000]
2. 1_{1000}2_{1000}3_{1000}4_{1000}5_{1000}6_{1000}7_{1000}8_{1000}9_{1000}0_{3339}1 12340
3. 1_{1}2_{2}3_{4}4_{8}5_{16}6_{32}7_{64}8_{128}9_{4220}  4475
4. 2_{2}3_{4}4_{8}5_{16}6_{32}7_{64}8_{128}9_{256}0_{3207}1  3718
5. 1_{111}2_{111}3_{111}4_{111}5_{111}6_{111}7_{111}8_{111}9_{111}0_{1917}1  2917
6. 1_{111}2_{111}3_{111}4_{111}5_{111}6_{111}7_{111}8_{111}9_{111}0_{1667}1  2667
7. 1_{1}2_{2}3_{4}4_{8}5_{16}6_{32}7_{64}8_{128}9_{2145}  2400
8. 1_{1}2_{2}3_{4}4_{8}5_{16}6_{32}7_{64}8_{128}9_{1866}  2121
9. 1_{50}2_{50}3_{50}4_{50}5_{50}6_{50}7_{50}8_{50}9_{50}0_{1255}1  1706
10. 1_{50}2_{50}3_{50}4_{50}5_{50}6_{50}7_{50}8_{50}9_{50}0_{1065}1  1516

* All discovered by H. Dubner Table 31: The TOP TEN Unholey Primes [ 1, 2, 3, 5, 7 ]

or  Digits
*1. [(17275727273727273727275727) · R_{3120}/R_{26}] · 10 + 1 (17275727273727273727275727)_{120}1 3121

HD:1992

**2. [(1(2)_{17}35553(2)_{17})(R_{3120}/R_{40})] · 10 + 1  (1(2)_{17}35553(2)_{17})_{78}1  3121
***3. (72323252323272325252)(10^{3120} − 1)/(10^{20} − 1) + 1  (72323252323272325252)_{156} + 1  3120
***4. (72323232723232525252)(10^{3120} − 1)/(10^{20} − 1) + 1  (72323232723232525252)_{156} + 1  3120
*5. 370 · ((100^{1441}) − 1)/99 + 3  (37)_{1441}3  2883
*6. [(173737573727572727572737573737)R_{2160}/R_{30}] · 10 + 1 (173737573727572727572737573737)_{72}1 2161
*7. [(173737572727375757372727573737)R_{2160}/R_{30}] · 10 + 1 (173737572727375757372727573737)_{72}1 2161
*8. [(173727375757572727575757372737)R_{2160}/R_{30}] · 10 + 1 (173727375757572727575757372737)_{72}1 2161
****9. (752275532)(10^{2160} − 1)/(10^{9} − 1) + 1  (752275532)_{240} + 1  2160
***10. (723232523232327272)(10^{2160} − 1)/(10^{18} − 1) + 1  (723232523232327272)_{120} + 1  2160
****10. (575253222)(10^{2160} − 1)/(10^{9} − 1) + 1  (575253222)_{240} + 1  2160
* Undulating and palindromic
** Palindromic
*** Undulating anti-palindromic with all prime digits
**** All prime digits

Table 32: The TOP TEN Prime Numbers with Prime Digits [ 2, 3, 5, 7 ]

or  Digits
*1. (72323252323272325252)(10^{3120} − 1)/(10^{20} − 1) + 1 (72323252323272325252)_{156} + 1 3120

HD:1992

*2. (72323232723232525252)(10^{3120} − 1)/(10^{20} − 1) + 1 (72323232723232525252)_{156} + 1 3120
**3. 370 · ((100^{1441}) − 1)/99 + 3  (37)_{1441}3  2883
4. (752275532)(10^{2160} − 1)/(10^{9} − 1) + 1  (752275532)_{240} + 1  2160
*5. (723232523232327272)(10^{2160} − 1)/(10^{18} − 1) + 1  (723232523232327272)_{120} + 1  2160
6. (575253222)(10^{2160} − 1)/(10^{9} − 1) + 1  (575253222)_{240} + 1  2160
*7. (523232525252327272)(10^{2160} − 1)/(10^{18} − 1) + 1  (523232525252327272)_{120} + 1  2160
*8. (523232325232525252)(10^{2160} − 1)/(10^{18} − 1) + 1  (523232325232525252)_{120} + 1  2160
***9. (323232727272327272)(10^{2160} − 1)/(10^{18} − 1) + 1  (323232727272327272)_{120} + 1  2160
10. (323222)(10^{2160} − 1)/(10^{6} − 1) + 1  (323222)_{360} + 1  2160
* Anti-palindromic with undulating digits
** Undulating digits and palindrome
*** Undulating digits

Table 33: The TOP TEN Lucas Prime Numbers

[ L_{n} = \underline{2}, 1, \underline{3}, 4, \underline{7}, \underline{11}, 18, \underline{29}, \underline{47}, …]

Digits

1. L(35449) 7909 [B. DeWater: 2001]
2. L(14449) 3020
3. L(10691) 2235
4. L(8467)  1770
5. L(7741)  1618
6. L(5851)  1223
7. L(4793)  1002
8. L(4787)  1001
9. L(1361)  285
10. L(1097)  230

[ and the rest ]

11. L(863) 181  24. L(37) 8
12. L(617) 129  25. L(31) 7
13. L(613) 129  26. L(19) 4
14. L(503) 106  27. L(17) 4
15. L(353) 74  28. L(16) 4
16. L(313) 66  29. L(13) 3
17. L(113) 24  30. L(11) 3
18. L(79)  17  31. L(8) 2
19. L(71)  15  32. L(7) 2
20. L(61)  13  33. L(5) 2
21. L(53)  12  34. L(4) 1
22. L(47)  10  35. L(2) 1
23. L(41)  9  36. L(0) 1

Table 34: The TOP TEN Prime Fibonacci Numbers

[F_{n} = 1, 1, \underline{2}, \underline{3}, \underline{5}, 8, \underline{13}, 21, 34, …]

Digits

1. F(9677) 2023  [B. deWater: 2000]
2. F(9311) 1946
3. F(5387) 1126
4. F(4723)  987
5. F(2971)  621
6. F(571)  119
7. F(569)  119
8. F(509)  107
9. F(449)  94
10. F(433)  91

[and the rest]

D  D
11. F(431)  90  19. F(23) 5
12. F(359)  75  20. F(17) 4
13. F(137)  29  21. F(13) 3
14. F(131)  28  22. F(11) 2
15. F(83)  17  23. F(7) 2
16. F(47)  10  24. F(5) 1
17. F(43)  9  25. F(4) 1
18. F(29)  6  26. F(3) 1

Table 35: The TOP TEN Countdown Prime Numbers [ 10 or 9 8 7 6 5 4 3 2 1 ]

Digits

1. 9876543210_{2002}1  2012 [C. Rivera: 1997]
2. (987654321)_{10}0_{1515}1  1606
3. (987654321)_{10}0_{1388}1  1479
4. (987654321)_{10}0_{1291}1  1382
5. (9876543210)_{10}0_{1222}1  1323
*6. (1098765432)_{131}1098765433  1320
7. (987654321)_{10}0_{979}1  1070
**8. (10987654321234567890)_{42}1  841
9. (9876543210)_{38}1  381
10. (9876543210)_{9}1  91
* Antipalindrome
** Almost-equi-pandigital palindrome

Table 36: The TOP TEN Reversible Primes (nonpalindromic)

[ 10_{xxx}“REV ERSE”0_{xxx}1 ]

Digits

1. 10_{850}20471010_{850}1  1709 [H. Dubner: 1997]
2. 10_{849}74408010_{849}1  1707
3. 10_{749}26149310_{749}1  1507
4. 10_{747}83012010_{747}1  1503
5. 10_{600}40544310_{600}1  1209
6. 10_{503}28886010_{503}1  1015
7. 10_{502}40982210_{502}1  1013
8. 10_{500}16338110_{500}1  1009
* 9. 10_{496}959260987654320_{495}1 1007
10. 10_{496}40100210_{496}1  1001

* Pandigital Table 37: The TOP TEN Primes with Straight Digits

[ 1, 4, 7 ] Digits

* 1. (14)_{815}1  1631 [C. Rivera: 1997]
** 2. 7_{1368}1  1369
** 3. 7_{1066}1  1067
*** 4. R_{1031} or 1_{1031} 1031
5. 17_{1001}1  1003
** 6. 7_{924}1  925
* 7. (1474)_{231}1  925
* 8. (14)_{291}1  583
** 9. 17_{510}  511
* 10. 74_{483}7  485
* Palindrome
** Near repdigit prime
*** Repunit palindrome

Table 38: The TOP TEN Primes with Largest Unique Periods

or  Period Digits
1. 10_{125}9_{249}89_{124}89_{125}0_{249}10_{124}1  3750  1001
* 2. (10^{1132} + 1)/10001  (99990000)_{141} + 1 2264  1128
3. (10_{12}9_{23}89_{12}0_{23})_{5}0(9_{23}89_{12}0_{23}10_{12})_{5} + 1  2232  721
* 4. (10^{922} + 1)/(101)  (9900)_{230} + 1  1844  920
5. (100999899000)_{25}0(999899000100)_{25} + 1  1812  601
6. (10_{4}9_{7}89_{4}0_{7})_{12}0(9_{7}89_{4}0_{7}10_{4})_{12} + 1  1752  577
* 7. (9_{39}0_{78})_{4}(9_{78}0_{39})_{4} + 1  1521  936
** 8. (10^{641} + 1)/11  (90)_{320} + 1  1282  640
* 9. (10^{586} + 1)/101  (9900)_{146} + 1  1172  584
*** 10. R_{1031}  1_{1031}  1031  1031

* Antipalindromic
** Undulating and antipalindromic
*** Largest known repunit

Table 39: The TOP TEN Absolute Prime Numbers

Digits

1. R_{1031} 1031 (H. C. Williams & H. Dubner: 1985]
2. R_{317}  317
3. R_{23}  23
4. R_{19}  19
* 5. 991  3
* 6. 919  3
* 7. 733  3
* 8. 373  3
* 9. 337  3
* 10. 311  3

* Actual Primes

[ and the rest ] Actual Primes

11. 199  19. 31
12. 131  20. 17
13. 113  21. 13
14. 97  22. 11
15. 79  23. 7
16. 73  24. 5
17. 71  25. 3
18. 37  26. 2

Table 40: the TOP TEN Consecutive Primes in Arithmetic Progression [ P, P+210, P+420, P+630, P+840, P+1050, P+1260, P+1470, P+1680, P+1890 ]

1. P + 1890 = 100 99697 24697 14247 63778 66555 87969 84032 95095 24689 19004

18036 03417 75890 43417 03348 88215 90672 31609

2. P + 1680 = 100 99697 . . . . .

. . . . . 31399.

3. P + 1470 = 100 99697 . . . . .

. . . . . 31189.

4. P + 1260 = 100 99697 . . . . .

. . . . . 30979.

5. P + 1050 = 100 99697 . . . . .

. . . . . 30769.

6. P + 840 = 100 99697 . . . . .

. . . . . 30559.

7. P + 630 = 100 99697 . . . . .

. . . . . 30349.

8. P + 420 = 100 99697 . . . . .

. . . . . 30139.

9. P + 210 = 100 99697 . . . . .

. . . . . 29929.

10. P  = 100 99697 24697 14247 63778 66555 87969 84032 95093 24589 19004

18036 03417 75890 43417 03348 88215 90672 29719

Discovered in 1998 by M. Toplic etal, these 93-digit primes have a common difference of 210.

Table 41: The TOP TEN Known Prime Factors of Googolplex Plus One

[Googolplex +1 = 10^{10^{100}} + 1 = 10^{100000000000000000000000000000000000−} 00000000000000000000000000000000000000000000000000000000000000000 + 1]

Digits

1. 370791604769783808 · 10^{63} + 1  81 [R. Harley: 1994]
2. 156941061512238345486336 · 10^{48} + 1  72
3. 353433491556781785088 · 10^{50} + 1  71
4. 325123864299130847232 · 10^{42} + 1  63
5. 83010348331692982272 · 10^{41} + 1  61
6. 8019958276735747672058735099904 · 10^{21} + 1  52
7. 493333612765059415097397477376 · 10^{21} + 1  51
8. 98231572083407610249634223245754368 · 10^{8} + 1 43
9. 73994152028165749791270453116928 · 10^{11} + 1  43
10. 4951760157141521099596496896 · 10^{14} + 1  42

Table 42: Ten Types of “Rare” Prime Numbers Number Known

A. Generalized Fermat Prime (or Depression Prime) of the type:  (1)

10^{2^{n}} + 1; n = 1.

B. Sequential prime of the type: (1234567890)_{n}1 ; n = 17, 56.  (2)
C. \underline{Almost-equi}pandigital prime of the type: (12345678901098765432)_{n}1;  (2)

n = 6, 16.

D. Type: n^{n^{n}} + 1; n = 1, 2.  (2)
E. Type: n^{n} + 1; n = 1, 2, 4.  (3)
F. Wilson primes (p): 5, 13, 563; (p − 1)! + 1 divisible by p^{2}.  (3)
G. Near repdigit type: 8_{n}9 ; n = 1, 13, 16, 34.  (4)
H. Fermat primes: 2^{2^{n}} + 1 ; n = 0, 1, 2, 3, 4.  (5)
I. Repunit primes: R(n): n = 2, 19, 23, 317, 1031.  (5)
J. Beastly palindrome of the the type: (10^{n} + 666) · 10^{n−2} + 1 ;  (7)

n = 3, 16, 45, 509, 611, 2475, 2626.