PART 1 (1 ~ 1,000,000,000,000)
The Single-Digit Range (1 ~ 9.9999999999999999)
1
1 is the building block of all other numbers, because every positive whole number can be expressed as 1 + 1 + ... + 1 + 1. It is the number such that n multiplied by that number is still n, which is known as the multiplicative identity. When plugged into googological functions, 1 often leads to degenerate cases, e. g. x*1 = x, x1 = x, x^^1 = x, x^^^....^^^^1 (with as many ^'s as we please) = x, and in the fast-growing hierarchy, if we plug 1 into any function, the function index will just tick down until we reach
1 + 1/G(64)
1 plus the reciprocal of Graham's number. This is a commonly cited example of an extremely small superuniary number. This number consists of 1.000000000...000000??????......788111111111...111111??????.........899222222222......222222??????......010333333333......333333???.........121444444......444444??????.........232555555555.........555555??????.........343666666666.........666666??????......454777777777......777777??????.........565888888888......888889??????.........677, then the whole decimal repeats. The number of 0s at the beginning (and the number of each digit in the long runs of repeated digits) is unfathomably large. The result of extracting the repeating part from this number and removing the leading 0s is r(G(64)) using the reptend function, or r(3, n) where 3^n = G(64).
1.1
1.2
1.3
1.41421356237309504880...
The square root of 2. It is sometimes referred to as "Pythagoras' constant".
1.455433238907...
The value of 2 tetrated to a half. Note that this is NOT the solution of xx = 2 (which is actually closer to 1.56), because, unlike with exponentiation, (a^^b)^^c =! a^^(b*c), and in fact is usually in the neighborhood of a^^(b+c-1).
1.61803398874989484820...
The golden ratio. It is the number x such that x+1 = x^2. It is the asymptotic ratio of two consecutive members of the Fibonacci sequence (or really any sequence where each term is the sum of the two before it), and can be expressed exactly as (sqrt(5)+1)/2.
2
2 is the first even number, and the first prime number. It is the only prime number that is even, because it divides all even numbers. It is the base of the binary system. 2+2=4, 2*2=4, 2^2 = 4, and the pattern continues into the hyperoperators: 2^^2 = 4, 2^^^2 = 4, and 2^^^...^^^2 with any number of arrows just collapses to 4.
2.50618414558876925629...
The solution to x^x = 10 (and also the solution to log10(x) = 1/x).
2.718281828459045235360287471352662...
The mathematical constant e. It is the sum of the reciprocals of the factorials, and also the base of the natural logarithm. It is the limit of (1 + 1/x)^x as x goes to infinity, so the first digits of (10^n + 1)^(10^n) (and (10^n + 1)^^2) converge to the digits of e.
3
3 is the sum of 1 and 2, the first odd prime, the only prime that is 1 less than a square number, and the only prime number that is triangular. It is also the only Mersenne number that is also a Fermat number (22 - 1 = 22^0 + 1). 3 is also the smallest number that does not lead to degenerate cases when plugged into the hyperoperators with itself as the other argument, so many googolisms are based on 3.
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384...
Perhaps the most famous mathematical constant of them all is pi, the ratio of the circumference of a circle to its diameter.
4
The first square number other than 1. Also the result of any hyperoperation with two 2s as the arguments, and thus the result of any BEAF array that begins with a 2 (because expansion iterates the up-arrows with the two arguments the same).
5
The only prime number with a 5 as the last digit. 5 is also half of our numeral base, so the multiples of 5 are somewhat round numbers.
6
2*3. If you multiply an even number by 6, the last digit stays the same, while if you multiply an odd number by 6, the last digit is incremented by 5.
7
Often considered a lucky number. There are 7 colors in the rainbow, and 7 days in the week.
8
Cube of 2.
9
The square of 3, and the only square to be of the form (10^n)-1.
The Double-Digit Range (10 ~ 99.999999999999999999)
10
The base of the decimal system.
11
12
13
7th Fibonacci number. Commonly associated with bad luck in Western culture.
14
15
16
The fourth power of 2, and also the square of 4. It is the only non-trivial integer solution of xy = yx, and it is also equal to both (2^2)^2 and 2^(2^2).
17
The third Fermat prime.
18
19
The only non-trivial number that is the sum of the consecutive products of its digits: 1*1 + 9*2 = 19 (There are no further such numbers because a 3 digit number is obviously in the hundreds, but the consecutive products of 3 digits can't add up to any more than 9*1+9*2+9*3 = 54). The first prime of the form 10^n - 9^n. Also the fourth alternating factorial.
20
21
6th triangular number, 8th Fibonacci number.
22
23
24
The factorial of 4, and the number of hours in a day.
25
Square of 5.
26
The number of letters in the English alphabet.
27
Third power of 3. Can also be expressed as 9*3, 3*9, or 3^^2.
28
7th triangular number.
29
30
31
32
Fifth power of 2.
33
34
9th Fibonacci number.
35
5th tetrahedral number.
36
Square of 6, and 8th triangular number.
37
The reptend in 1/27, expressible as r(3, 3) using the reptend function. It is the only nontrivial prime number of the form (103^(n-2) - 1)/3n (or r(3, n)), as it divides all further such numbers.
38
39
3*13, and the sum of 3 + 5 + 7 + 11 + 13.
40
41
Part of a formula (x2 + x + 41) that generates a prime for any number greater than -40 and less than 40. It fails at x=40: plugging in 40 gives 402+40+41, which is equal to 402+80+1, which is equal to 412 since (x+1)2 = x2 + 2x + 1 = x2 + x + (x + 1).
42
The answer to the Ultimate Question in Hitchhiker's Guide to the Galaxy.
43
The smallest non-trivial ascending power number (42 + 33 = 43). Also a member of the sequence 1, 2, 3, 7, 43, 1807... where each term is the product of all the terms before it, plus one. This is another sequence that can be used to show that there are infinitely many primes, because a term in this sequence will leave remainder 1 when divided by any of the previous numbers in the sequence.
44
45
The sum of all decimal digits. Because 45 is 5*9, a number with an equal number of instances of every digit is divisible by 9.
A Kaprekar number is a number such that if you take the square and add up the blocks of digits with as many digits as the original number, you get the original number again. 452 is 2025, and 20 + 25 = 45. In fact, the blocks of 2 digits of 453 and 454 also add up to 45: (0)9 + 11 +25 = 45, and 4 + 10 + (0)6 + 25 = 45, and it is the only known number that is a Kaprekar number for squares, cubes, and fourth powers.
46
47
48
49
Square of 7, and its reciprocal includes the powers of 2 until they start to overlap and break the pattern: 0.02040816326530......
50
51
52
The number of playing cards in a deck.
53
54
55
The sum of all whole numbers from 1 to 10, and also a Fibonacci number (the 10th Fibonacci number, counting both of the initial 1s), and 5+5=10. The largest number that is a triangular number and a Fibonacci number.
56
57
58
59
60
The number of minutes in an hour, and the number of seconds in a minute. The Babylonians used 60 as the base in their number system because 60 can be divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30.
61
A(3, 3), where A denotes the Ackermann function, and thus equal to gag(3). We will see the next gag- number in part 3. With a degree of 3, the Ackermann function does not give exactly the powers of 2, and instead gives 2^(n+3)-3, which is somewhat reminiscent of the Mersenne numbers. Many of the first values of A(3, n) are prime.
62
63
64
26. Can also be expressed as 43, or 82. The first of only two powers of 2 with more than one digit that have only even digits.
65
66
The sum of all whole numbers from 1 to 11.
67
68
69
70
71
72
73
The "Chuck Norris of numbers": 73 = 21st prime, 37 = 12th prime.
74
75
76
77
78
The sum of all whole numbers from 1 to 12.
79
80
81
The square of 9, and the fourth power of 3. Can also be expressed as 3^2^2.
82
83
84
85
86
87
88
89
A Fibonacci prime, and also the smallest non-trivial ascending power number with the powers starting at 1 (81 + 92 = 89)
90
91
92
93
94
95
96
97
98
99
One more than twice a square (72*2 + 1) and one less than 100.
The Hundreds Range (100 ~ 1000)
100
101
The largest known prime of the form 10n + 1. It can be shown that for all primes of the form nk + 1, k must be a power of 2 regardless of n, because n + 1 divides all number of the form n2k + 1 + 1, and n2 + 1 divides all numbers of the form n4k + 2 + 1, and so on.
105
108
111
The smallest repunit divisible by 3: 111 = 3*37.
120
The factorial of 5
121
The square of 11.
125
The cube of 5. Also the third Friedman number because 125 = 51+2.
127
27 - 1, fourth Mersenne prime, and the third Catalan-Mersenne number. The largest number that can be stored in a signed byte.
Also has an interesting pattern in its reciprocal: 0.007874 015748 031496 062992 125984..., as 127 divides 999,998.
As in most cases where such patterns appear, 7874 is itself divisible by 127, so the pattern continues into 1/16,129.
128
Seventh power of 2.
135
An ascending power number (11 + 32 + 53 = 135).
136
A triangular number that is formed by the concatenation of the first 3 triangular numbers.
137
Approximately the reciprocal of the fine structure constant.
144
Square of 12, or a dozen dozen (a gross).
150
169
180
196
200
216
Cube of 6.
225
243
35. This number sounds random in decimal, but not in ternary.
250
256
28. Can also be expressed as 44, 162, 4^^2 or 22^3. The number of different values that can be represented in one byte.
257
Fourth Fermat prime.
270
271
r(369) using the reptend function, an example of a value of the reptend function that is surprisingly small (because 1/41 has a period of just 5). The second prime of the form 10^n - 9^n.
289
300
324
332
A descending power number: 35 + 34 + 23 = 332. A descending power number is a number where the sum of the descending powers of the digits starting from a certain number is equal to the original number.
341
The first super-Poulet number. A super-Poulet number is a number where each of the divisors of the number, d, divides 2d - 2 (and thus 2d - 1 - 1).
343
The cube of 7.
360
The number of degrees in a circle. 360 can be evenly divided by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 45, 60, 72, 90, 120, and 180.
361
Square of 19, and also has an interesting pattern in its reciprocal: 0.00277 00831 02493 07479 22437 67313 01939 ......
365.2425
The average number of days in a Gregorian year. There are 365 days in a normal year, and 24.25% of years are leap years.
400
403
Entry forbidden
404
Error: entry not found
A number with a more immediate meaning on the Internet, because it is the HTTP status code for "Not Found", one of the most common errors. There will be two more joke "error entries".
441
Square of 21.
484
500
Half of one thousand. D in Roman numerals. Also the most common error code given by a website that has crashed.
503
Error: entry (temporarily) unavailable
Another common error code indicating a website is down.
512
29. Can also be expressed as 83. 512 is also a Dudeney number, because 5 + 1 + 2 = 8.
561
The smallest Carmichael number.
619
The sixth alternating factorial.
625
Fourth power of 5.
666
The famed "number of the beast". The sum of all numbers from 1 to 36 (which happens to be 62), and also the sum of 13 + 23 + 33 + 43 + 53 + 63 + 53 + 43 + 33 + 23 + 13. Also the sum of the squares of the first 7 primes: 22 + 32 + 52 + 72 + 112 + 132 + 172 = 666.
There are several theories on why 666 is the number of the beast. One theory links it to the Roman emperor Nero, who the early Christians would have considered evil.
676
The square of 26. The first palindromic square whose square root is not a palindrome.
700
720
The factorial of 6.
729
745
When I was very young, I would assign numbers to myself and people around me. 745 was the number I gave to myself, which meant I would use it in user account names after getting my first computer.
777
841
The square of 29, and the sum of the squares of 20 and 21. Thus, a right triangle with side lengths 20 and 21 will have a hypotenuse with length 29. This is called a Pythagorean triple.
945
997
Most notable for an interesting tripling-and-shifting pattern in its reciprocal:
0.001 003 009 027 081 243 731 193 580 742 226 680 040 120 361 083 ...
The pattern occurs because 1000/997 = 1 + 3/997 = 1 + .003 + .009/997, and so on.
998
999
The Two-Block Range (1,000 ~ 1,000,000)
1,001
sphenic numbers
A sphenic number is a number that is a product of three primes. 1001 is equal to 7*11*13, so it is also the product of three consecutive primes.
1,024
210. Can also be expressed as 45 or 322. A perfect power that is close to a power of 10. The number of bytes in a kilobyte.
1,111
repunits
A repunit is a number consisting of only ones, expressible as (10^n - 1)/9. 1111 is an example of a repunit. It is a composite one, being equal to 11*101.
1,387
The second super-Poulet number.
1,676
A number that is both an ascending power number and a descending power number: 11 + 62 + 73 + 64 and 15 + 64 + 73 + 62 both equal 1,676.
1,729
The first number that can be expressed as the sum of cubes in two different ways: 93 + 103 = 13 + 123. Also a near miss to Fermat's Last Theorem.
1,961
The most recent strobogrammatic year, and the last one until 6009 (3991 years from now).
2,000
2,012
The most recent year to be both left and right polydivisible. The year began on a Sunday (like 2017 and 2023) and was a leap year.
2,014
The second of three consecutive sphenic year numbers, and the year I first became interested in large numbers. The year began on a Wednesday.
2,016
The number of the year I created this site, also the recent year that was divisible in the most ways: 2016 is divisible by all numbers from 1 to 9 except 5, as well as 12, 14, 16, 18, 24, 28, 32, 42, 48, 63, 126, 252, 504, and 1008, for a total of 36 divisors (including 1 and itself, and even the number of divisors). Also, 2016*5/2 is equal to 7!. The year began on Friday and was a leap year.
2,017
The number of board-pile polyominoes with 8 cells. The year began on a Sunday, and the weekdays of dates beginning in March had not happened since 2006.
2,018
The sum of all squares from 72 to 182, the sum of the fourth powers of 2, 3, 5, and 6.(and the year I first joined the Googology Wiki). Also the sum of 132 and 432. The year began on a Monday.
2,019
The year toward the end of which my interest in numbers really began to rise again. The year began on a Tuesday, and thus had all days the same weekday as in 2013.
2,020
A self-descriptive number if we loosen the requirement that the number of digits be the base (which in decimal is 10, of course). The year began on Wednesday and was a leap year.
2,038
The number of the year in which Unix time will reach 2,147,483,648. See 2147483647 on the next page for more. The year will begin on a Friday.
2,047
The first number of the form 2p - 1 (where p is a prime number) that is not prime. It is equal to 23*89.
2,048
211. The largest (known) power of 2 where all digits are even. However, there are larger powers of 2 that end with several even digits (see the next page). The year will have the same weekday of every date as 2020.
2,187
The seventh power of 3. The nine's complement of its reverse: 2187 + 7812 = 9999.
2,380
2,401
Fourth power of 7. Its digits also add up to 7, meaning it is a Dudeney number for 4th powers. See 512 and 19,683.
2,500
2,
2,916
3,000
3,125
The fifth power of 5. Can also be expressed as 5^^2.
3,249
Square of 57, and the first number whose factorial is greater than 10^10000.
3,435
A perfect digit-to-digit invariant: 3^3 + 4^4 + 3^3 + 5^5 = 3435.
3,439
The first value of 10^n - 9^n that is not prime: 3439 = 19*181.
3,713.155056406...
The value of "megafuga-e".
4,000
4,096
212. Can also be expressed as 46, 84, 163, and 642. Also, 4096 is its first digit to the power of its last digit. The next number of the form ab that starts with a and ends in b is 387, on the next page.
4,098
The lower bound for the fifth Busy Beaver number.
4,150
A perfect digital invariant that is the sum of the fifth powers of its digits.
4,151
Another perfect digital invariant that is the sum of the fifth powers of its digits, comprising a pair with 4,150.
4,369
The seventh Super-Poulet number.
4,421
The seventh alternating factorial.
4,503
The largest number that is not the sum of four or fewer squares of composite numbers, and the first 4 digits of 252.
4,655
The number of decominoes, which are polyominoes made from 10 squares.
4,913
Cube of 17, and a Dudeney number: 4+9+1+3 = 17.
5,000
5,040
The factorial of 7 (1*2*3*4*5*6*7), and the largest factorial that is a highly composite number. Also equivalent to 10*9*8*7.
5,041
712. Notable for being one more than 7!. Also equal to 7! + 1!, and thus the sum of the factorials of the digits of its square root.
5,120
5*45. Can also be expressed as 5*210. The latter expression uses the digits of 5120 in a different order.
5,280
The number of feet in a mile.
5,760
Has 48 divisors, including the number of divisors it has.
5,814
17*18*19, and the first 4 digits of 354.
6,561
38. Can also be expressed as 94, or 812.
7,560
The first number that is highly composite but not superabundant.
7,776
Fifth power of 6. It is easy to remember mostly because it is almost a repdigit.
8,191
213 - 1, a Mersenne prime. It is the largest number that is a repunit prime in more than one base: 8191 = 11111111111112 = 11190.
8,192
Thirteenth power of 2.
9,000
A number commonly referenced in the Over 9000 meme.
9,001
The first whole number that is actually over 9000, and thus also referenced in the Over 9000 meme. There was one video where someone tried to divide by zero and 9001 appeared in the output box (an obvious cut). We will get to what the Internet means by "over 9000" later in this list.
10,000
Ten thousand.
10,001
The first number of the form 102^n + 1 that is not prime (it is equal to 73*137). There are no more generalized Fermat primes of the form 10^2^n + 1 up to 10^16777216 + 1.
10,958
The first number for which there is no known way to express the number using the digits 1, 2, 3, 4, 5, 6, 7, 8, and 9 in that order and the operations of addition, multiplication, and subtraction. Also, there are sometimes this many days in exactly 30 years.
12,321
Square of 111.
14,641
114, the highest power of 11 whose digits coincide with rows of Pascal's triangle, and also the highest one that is palindromic.
16,384
214. Can also be expressed as 47.
19,683
39. Can also be expressed as 273, or 3^3^2. This number is equal to fuga(3).
20,736
Fourth power of 12.
27,648
The hyperfactorial of 4: 11*22*33*44 = 27648.
32,768
215. Can also be expressed as 85 or 323.
34,560
The superfactorial of 5: 1!*2!*3!*4!*5! = 34560.
35,899
The eighth alternating factorial.
46,656
The sixth power of 6. Can also be expressed as 6^^2.
50,000
59,049
Tenth power of 3.
65,536
216. Can also be expressed as 2^2^4, 2^2^2^2, 2^^4, and 2^^^3. The number of values that can be stored in a 16-bit space.
65,537
The largest known Fermat prime.
86,400
The number of seconds in a day.
100,000
104,976
130,321
131,071
131,072
142,857
(106-1)/7, r(7) using the reptend function. This number has the property that if you multiply it by any integer less than 7, you get a cyclic permutation of its digits: 142857*2 = 285714, 142857*3 = 428571, 142857*4 = 571428, 142857*5 = 714285, and 142857*6 = 857142.
146,097
The number of days in 400 years (a full Gregorian calendar cycle). It is equal to 7*27*773.
232,324
Square of 482. A 6-digit square number that is particularly easy to remember, since it is one more than 481*483 = 10101*23.
262,144
218. Can also be expressed as 49, 86, 643, or 5122. Also equal to 4^3^2^1, and thus the fourth exponential factorial. The next exponential factorial is a number of 183,231 digits.
275,625
Square of 525. The square of the first three digits is the original number with the first digit removed, and this can be repeated again:
275625 = 5252, 2752 = 75625,
75625 = 2752, 752 = 5625
Also, 275625 remains square if the first digit is removed repeatedly until just the 5 is left, which is not square:
275625 = 5252
75625 = 2752
5625 = 752
625 = 252
25 = 52
5
275625 is also the third in a sequence of square numbers formed by substituting the square root in the thousands starting with 5625, which can be continued for one more term: 525625 = 7252.
333,667
If you multiply a 3-digit number by 3, and then by 333667, you get the original 3 digit number repeated exactly three times. This works because 333667 x 3 = 1001001.
362,880
The factorial of 9.
500,000
524,287
The seventh Mersenne prime, equal to 219 - 1.
823,543
7^7. Can also be expressed as 7^^2.
998,001
9992. Also has a curious reciprocal: 1/998001 = 0.000 001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028....
The Three-Block Range (1,000,000 ~ 1,000,000,000)
1,000,000
One million.
1,048,576
220. Can also be expressed as 410, 165.
1,235,894
A number that appeared in one divide-by-zero video I remember.
1,594,323
313.
2,097,152
221. Can also be expressed as 87, or 1283.
2,424,249
Square of 1,557. 24 is repeated three times, because 1554*1560 = 777*13*2*120 = 10101*240.
3,612,703
A divisor of every alternating factorial starting with af(3612702). Thus there are a finite number of alternating factorial primes.
4,194,304
5,000,211
Cube of 171, only slightly more than 5 million.
5,217,031
The third prime of the form 10^n - 9^n.
5,764,801
78. Can also be expressed as 494.
8,388,608
223. 0.5 more than the largest non-whole number that can be expressed in the IEEE single-precision floating point format.
8,675,309
9,765,625
Tenth power of 5.
10,000,000
107, a pipsqueak.
10,077,696
69, a perfect power that is very close to a power of 10 (this time even closer to 10,000,000 than 210 is to 1000 and 223 is to 10000). Also a member of the sequence starting with 1, 2, and every term after that is the (n-2)th hyperoperation applied to all the terms in the sequence:
1, 2, 3, 6, 10077696, (a power tower of as many 10077696s as a power tower of 27 6s), ...
12,345,679
(109 - 1)/81, and thus the reptend in 1/81 (without the leading zero). Can be factored as 37 * 333667. It is the ninth number of the form (bb-1 - 1)/(b-1)2, which in base b would be nearly a pandigital number (with only the digit b-2 missing):
Quaternary: (43 - 1)/32 = 63/9 = 710 = 134
Base 5: (54 - 1)/42 = 624/16 = 3910 = 1245
Base 6: (65 - 1)/52 = 7775/25 = 31110 = 12356
Base 7: (76 - 1)/62 = 117648/36 = 326810 = 123467
Octal: (87 - 1)/72 = 2097151/49 = 4279910 = 1234578
Nonal: (98 - 1)/82 = 43046720/64 = 67260510 = 12345689
Hexadecimal: (1615 - 1)/152 = 1152921504606846975/225 = 512409557603043110 = 123456789ABCDF16
12,550,821
In very old versions of the sandbox game Minecraft, the "Far Lands" would initiate this many blocks from the world origin (and still existed in the mobile version of the game until relatively recently), due to the coordinates of two of the noise maps used in generating the terrain which increase by 684.412 every four-block sampling reaching 2^31. When beyond this number on one axis, large tunnels and stretched terrain would generate, while stacked land would generate past this point on both X and Z axes. I have extensively discussed the Far Lands on my YouTube channel. See also 2,147,483,647.
We will encounter more Minecraft-related numbers in this list.
14,348,907
315. Can also be expressed as 275 or 2433.
16,777,216
224. Can also be expressed as 412, 88, 8^^2, 166, 644, 2563, and 4,0962. The number of different colors in the RGB color format used by computers. Also one more than the largest odd number representable in the single-precision floating point format, which is the cause of a graphical glitch in Minecraft Bedrock Edition known as the Stripe Lands.
19,487,171
Seventh power of 11.
30,000,000
31,536,000
The number of seconds in a regular year.
32,000,000
33,554,432
225. Can also be expressed as 325. The number nearly consists of 4 consecutive double digits.
43,046,721
316.
50,000,000
51,151,104
Square of 7,152. 511 is repeated twice because 7150*7154 = 143*50*7*1022 = 143*7*51100 = 1001*511*100.
65,805,519
67,108,864
73,939,133
The largest left-truncatable prime.
80,802,121
Square of 8989.
87,539,319
The second taxicab number.
100,000,000
123,456,789
The first zeroless pandigital number. Multiplying 123456789 by any single digit integer not divisible by 3 gives a different permutation of the digits:
123456789 x 2 = 246913578
123456789 x 4 = 493827156
123456789 x 5 = 617283945
123456789 x 7 = 864197523
123456789 x 8 = 987654312
214,358,881
118. The first power of 11 to begin with a 2.
381,654,729
A polydivisible number that has all digits except zero. 3 is (obviously) divisible by 1, 38 is divisible by 2, 381 is divisible by 3, 3816 is divisible by 4, 38165 is divisible by 5, 381654 is divisible by 6, 3816547 is divisible by 7, 38165472 is divisible by 8, and 381654729 itself is divisible by 9
387,420,489
99. Can also be expressed as 9^^2, 318, or 276.
438,579,088
A perfect digit-to-digit invariant if we loosen the definition so that the 0 is not included. It is equal to the sums of the nonzero digits raised to the power of themselves.
479,001,600
The factorial of 12.
500,000,000
536,870,912
2^29, the largest power of 2 without redundant digits.
909,090,909
(1010 - 1)/11. The first number of the form (10p - 1 - 1)/p that is not a cyclic number.
987,654,321
123456789 in reverse. Also equivalent to 379721*512 or (8*1010+1)/81. This number even has the "123456789-property" when assuming that there is a leading zero:
0987654321 x 2 = 1975308642
0987654321 x 4 = 3950617284
0987654321 x 5 = 4938271605
0987654321 x 7 = 6913580247
0987654321 x 8 = 7901234568
The Four-Block Range (1,000,000,000 ~ 1,000,000,000,000)
1,000,000,000
One billion.
1,004,065,811
Approximately the coordinate in Minecraft worlds where a second set of Far Lands known as the Farther Lands would appear, which appear similar to the first set but more stretched out.
1,008,910,965
The hexatrigesimal "GOOGOL", which is of course nowhere close to a googol. See also 1717162949.
1,073,741,824
230. Can also be expressed as 415, 810, 326, 645, or 10243.
1,163,962,800
The first number that is superabundant but not highly composite.
1,717,162,949
The number that reads, "SECRET" in base 36. Since the hexatrigesimal (base 36) system uses all letters in the English alphabet, it is great for secret messages in the form of numbers. See also 1008910965.
2,147,483,647
The largest number that can be represented in a signed 32-bit space, and as such, the hard limit in many games (and other computer programs) which use 32-bit integers (including the old versions of Minecraft before a true hard boundary for worlds was implemented).
3,486,784,401
320. Can also be expressed as 910, 2434, or 590492.
4,294,967,295
232 - 1. The product of all known Fermat primes (3*5*17*257*65537).
4,294,967,296
232, Can also be expressed as 416, 168, 2564, 655362, or 2^(2^5). This is the number of values that a 32-bit space can hold.
4,294,967,297
The first Fermat number that is not prime: 4,294,967,297 = 641*6,700,417. All further Fermat numbers up to 2^2^32 + 1, which has more than 1.29 billion digits, are known to be composite.
4,341,201,053
Integer part of e^^pi using one definition of real-valued tetration, which is about 4341201053.37.
6,227,020,800
The factorial of 13.
8,000,000,000
A rough estimate of the current population of the world.
8,589,934,592
233.
10,000,000,000
1010. Can also be expressed as 10^^2, thus the name dialogue.
10,662,526,601
Cube of 2201. The largest known nontrivial palindromic cube.
17,179,869,184
234. The number of cells in an Excel spreadsheet.
34,359,738,368
2^35. In versions of Minecraft that were not modded to support terrain generation past 2^31, chunk coordinates would overflow this many blocks out, and attempting to approach this boundary or teleport past it would freeze the game.
36,363,636,364
The square of this number (which is in Part 2) is another 11-digit number concatenated with itself. This is related to 100,000,000,001 having 2 factors of 11.
76,923,076,923
The fourth number of the form (10p-1 - 1)/p where p is a prime other than 2 or 5. It is the second such integer that is not a cyclic number.
87,178,291,200
The factorial of 14.
100,000,000,000
137,438,691,328
The sixth perfect number.
165,165,836,836
Square of 406406.
285,311,670,611
1111.
456,456,979,456
Square of 675,616. All blocks of three are identical except for the thousands.
549,755,813,888
239. Can also be expressed as 813, or 81923. It is particularly notable because it ends in three consecutive 8s, which is the only way a power of 2 can end in three of the same digit.
>> PART 2