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622080000000000 = 2×100×360×2×1000×4320000

The length of the cycle of existence in Hindu mythology. After each mahayuga (a period of four ages lasting 4320000 years, see 432000) there is a period of another mahayuga during which the world is undifferentiated. 1000 mahayuga make up the kalpa or manvantara, 4320000000 years, called a "day of Brahma", 1 day in the life of the god Brahma. After such a period Brahma sleeps for another equal period, so the day and night of Brahma last 8640000000 years. 360 such day-night periods is the year of Brahma, 3110400000000 of our years (360 × 8640000000). 100 such years is the maha-kalpa, the lifetime of Brahma, 311040000000000 of our years. At this point everything including Brahma continues to exist for an equal period, then the cycle repeats over and over forever. The length of this cycle is 622080000000000 years.

758083947856951 =~ 7.580×10¹⁴

A "Cunningham chain" is a sequence of prime numbers where each one is twice the previous one plus one. For example, {2, 5, 11, 23, 47} is the first chain of length 5 and {89, 179, 359, 719, 1439, 2879} is the first of length 6. Gunter Loh found the first chain of length 13, beginning with 758083947856951.

1000000000000000 = 10¹⁵

10¹⁵ is a quadrillion in the United States. In France and under the old British system, it is called thousand billion (or sometimes billiard, no relation to the game). Colloquially people use the word zillion to refer to numbers larger than those with familiar names, that is, larger than trillion and therefore this is about the smallest "zillion".

1693182318746372

The first of a set of 1131 consecutive composite numbers: Every number from 1693182318746372 through 1693182318747502 is composite. This is the first time there are 1000 or more composite numbers in a row. See also 370262.

9460730472580800 = 9.4607304725808×10¹⁵

The distance (in meters) that light travels in a year, called a light-year. As recommended by the IAU, it is based on the official (defined) speed of light, and the Julian year. Because both of those have precise definitions, the light year is exactly the integer shown here.

See also 149597870691.

10000000000000000 = 10¹⁶

According to Richard Crandall⁶⁷, the "longest calculation ever performed" to get a yes-or-no answer, is 10¹⁶ fundamental operations, to prove that the 22^nd Fermat number (2²²²+1) is composite. They didn't even find any factors — merely a yes-or-no answer that there is a factor. Crandall says this amount of computation is equivalent to that used to produce the entire Pixar-Disney movie Toy Story.

100000000000000000 = 10¹⁷

An estimate of the number of words printed on paper during the first 500 years of the printing press (roughly 1456 to 1956). (From Straight Dope)

262537412640768743.9999999999992500725971981... = e^π√163 ≅ 2.625×10¹⁷

An example of a "coincidence" that isn't. The value of e^π√n is a near-integer for a few special values of n. There are twelve 9's in a row. The integer approximated by this is 262537412640768744, which is also equal to 640320³+744 =(2⁶×3×5×23×29)³ +(2³×3×31). The near equality can also be transformed into an approximation of π:

π ≅ ln(640320³+744) / √163

which is correct to 30 digits after the decimal point.

One constant like this is a fluke and almost believable if you look at it statistically, but it becomes more interesting when you know that there are a few other similar, smaller constants:

e^π√67=147197952743.999998662..., approximately equal to 5280^3 +744 =(2⁵×3×5×11)³+744
e^π√43=884736743.9997774..., approximately equal to 960^3 +744 =(2⁶×3×5)³+744
e^π√19=885479.7776..., approximately equal to 96^3 +744 =(2⁵×3)³+744

e^π√163 is sometimes called the "Ramanujan constant", but that is not historically accurate — there is no record of Ramanujan discussing this number, although he did discuss these, which do not fit into the same pattern of 744 plus a cube:

e^π√58=24591257751.9999998222...,
e^π√37=199148647.99997804....,
e^π√22=2508951.998257...,

There are far more than the statistically-expected number of occurrences of e^π√N falling within 1/100 of an integer for integer N: 11 occurrences for N<100 and 37 occurrences for N<1000. The first 11 all fall just below an integer value (giving two 9's after the decimal point), and then we start getting a mix of just-below and just-above values. See Sloane's A019296, or here for a list.

You have to get into some very deep mathematics to show why these "almost integers" occur, but the end result is that for certain values of N, if A=e^π√N),

A - 744 + 196884/A - 21493760/A² + 864299970/A³ - ... = B³

where B is some large integer. If A is large enough, then all the terms after the first two are real small and that means that A has to be real close to an integer.

Amazingly, the above expression and its coefficients are related to group theory and Fermat's Last Theorem. The coefficients are Sloane's sequence A000521. All of them (except for 744) are related to terms in Sloane's sequence A001379, the degrees of irreducible representations of Monster group M: 196884=196883+1; 21493760=21296876+196883+1; 864299970=842609326+21296876+2×(196883+1); ... See also 196883.

10¹⁸

An upper bound on the processing rate embodied by the human brain, based on 10¹¹ neurons with 10000 dentrites each and a firing rate of 1000 per second. A brute-force simulation of this could be done with about 10¹⁸ floating-point operations per second, although in practice an optimized simulation would probably be used instead, reducing that number to about 10¹⁴. See also 10⁹.

10¹⁹

An estimate of the total number of insects living in the world.⁶¹

See also 6788516573, 75250000000, 10¹⁴ and 5×10³⁰.

12157692622039623539 ≅ 1.216×10¹⁹

12157692622039623539 = 1¹ + 2² + 1³ + 5⁴ + ... + 3¹⁹ + 9²⁰, the sum of its own digits raised to consecutive powers. See 135.

18446744073709551615 = 2⁶⁴ - 1 ≅ 1.8447×10¹⁹

This number has two legends associated with it.

The first legend is ancient and concerns the origin of the game of chess. King Shirham of India offered the inventor of the game any reward he cared to name, and the inventor asked for wheat: 1 grain for the first square on the chessboard, 2 grains for the second, 4 for the third, 8 for the fourth and so on — doubling each time. The king was surprised, thinking this request was almost insultingly modest (thinking it would only add up to a few bushels), but obliged and instructed his servants to measure out the inventor's reward. They spent some time trying to do this, and eventually realized that all the wheat in all the kingdoms in the world would not come close to the amount specified. The number of grains of wheat requested is 2⁶⁴ - 1. (That is over 400 times the world's total wheat production in the year 2003, assuming a grain of wheat is 20 cubic millimeters.)

The second legend was created by Edward Lucas, the inventor of the Tower of Hanoi game, in 1884. As this story goes, at the center of the world (or, in another version, at the top of a mountain in the Himalayas), is a set of three rods bearing golden discs. There are 64 discs, and they are all different sizes. When the world was created, they were stacked on one of the rods, with the largest at the bottom and the smallest at the top. Ever since then, the priests have worked continually transferring the discs from one rod to another. They can only move one disc at a time, and they are not allowed ever to put a larger disc on top of a smaller disc. When they succeed in stacking all 64 discs on the second rod in a single stack as they were when the world was created, the world will end. The number of steps required to carry out this process is 2⁶⁴ - 1.

The fact that the number comes up in these two different stories seems even more of a coincidence when you consider that in the chess legend, the number is the sum 1+2+2²+2³+...+2⁶²+2⁶³, whereas in the Lucas legend it is 2⁶⁴-1. These happen to be the same sum, but arrived at in two different ways.

18446744073709551616 = 2⁶⁴ ≅ 1.8447×10¹⁹

This is 2⁶⁴ = 2²⁶ = (((((2²)²)²)²)²)².

The hyper4 operators

There are two ways to define an operator that follows next after addition, multiplication and exponents. Both definitions specify a function that takes two numbers A and B, and whose value is

A ^ A ^ A ^ ... ^ A

where the number of A's is B. But there are two obvious ways to define this, depending on whether you evaluate it from right to left or from left to right:

((..((A^A)^A)^...)^A)^A  or  A^(A^(A^...^(A^A)..))

If you group the parentheses from the right, you get the hyper4 operator, which is generally accepted as a better definition. If you group the parentheses from the left, you get something that grows much more slowly, but still pretty fast, and which I studied quite a bit in high school (late 1970's). I used the somewhat confusing name "powerlog" to refer to the function, as in "2 powerlog 7 equals 2⁶⁴". If you denote the operation by "_④", then

A_④B = (((A ^ A) ^ A) ^ ... ) ^ A = A^A(B-1)

One reason the hyper4 definition is chosen over this one is that hyper4 cannot be reduced to a simple formula like this. Like each of the functions before it, hyper4 cannot be defined as a simple finite combination of other "lower" functions.

Extending hyper4 to the reals

I have been asked several times whether the hyper4 operator can be extended to the reals (to compute, for example, π^④e). This issue is discussed at length here.

As discussed above, the lower-valued hyper4 operator can be expressed in terms of a double exponent: A_④B = A^A(B-1) — so it is easily extended to reals and complex numbers. For example, see 4979.003621.

There is more on the hyper4 and higher hyper operators here.

(2⁶⁴ also happens to be the largest integer I have ever memorized, except for trivial ones like vigintillion.)

43252003274489856000 = 8!×3⁸×12!×2¹²/(3×2×2) ≅ 4.3252×10¹⁹

This is the number of combinations of the 3×3×3 Rubik's Cube. The 8!×3⁸ term reflects shuffling and rotating the corner pieces; 12!×2¹² is for the edges; and the denominator 3×2×2 reflects the three constraints: the total rotation of the corners is always 360^o, the total rotation of the edges is also always 360^o, and you cannot perform a single swap {A,B}->{B,A} (if any one piece is out of place, at least two others must also be out of place). If the constraints are avoided by physically disassembling the cube, you get 519024039293878272000.

This number became a rather well-known example of "innumeracy", when the packaging of the puzzle (as sold in the United States by Ideal Toy Corporation) said that the puzzle had "over 3 billion combinations". While not technically wrong, it is quite an understatement to say the least — the actual number of combinations is greater than the square of 3 billion. Apparently, the company thought that customers would not understand "over 43 quintillion" quite as well as "over 3 billion".

154345556085770649600

This is the smallest 6-perfect number. Its divisors add up to exactly 6 times itself. Its prime factorization is: 2¹⁵×3⁵×5²×7²×11×13×17×19×31×43×257. This is the smallest case of a K-perfect number where K itself is a perfect number (-:

519024039293878272000 = 8!×3⁸×12!×2¹² ≅ 5.1902×10²⁰

This is the number of combinations of the 3×3×3 Rubik's Cube if you're allowed to take it apart and put it back together. See also 43252003274489856000 and 101097362223624462291180422369532000000.

6670903752021072936960 ~= 6.67×10²¹

This is the number of ways of arranging the digits 1 through 9 in a 9×9 grid so that each of the 9 mutually disjoint 3×3 subsquares contains each digit once (such grids of numbers are used in the popular sudoku puzzles). The number has been calculated and verified by two independent researchers using different exhaustive counting (brute-force search) algorithms³³. See also 5524751496156892842531225600.

31858749840007945920321 = 422481⁴ ≅ 3.1859×10²²

This is 422481⁴, equal to 95800⁴ +217519⁴ +414560⁴. This is the simplest known counterexample to the conjecture of Euler that an n^th power cannot be expressed as the sum of less than n smaller n^th powers. In this case it's the sum of 3 4^th powers. Of course, it can never happen with a sum of two 3^rd powers, because that would be a counterexample to Fermat's Last Theorem. See also 216.

70000000000000000000000 = 7×10²²

This is the approximate number of stars in the universe, based on the strip survey estimate of Dr. Simon Driver's team, announced in early 2004. The previous estimate of 3.2×10²² had been established based on observations by the Hubble telescope. These numbers are really hard to estimate because astronomers cannot see most of the stars in our galaxy (due to clouds of dust, and because most stars are too faint) or most of the galaxies (same reasons) so it ends up being the product of a lot of statistical guess-work. Also, the definitions of "star" and "universe" are the subjects of constant debate. See also 1×10⁹⁷.

278914005382139703576000 ≅ 2.789×10²³

This is 2⁶×3⁵×5³×7²×11×13×17×19×23×29×31×37×41×43×47, and is the smallest number with over a million distinct factors (1032192 to be exact). See also 12, 840, 45360, 720720, 3603600, 245044800, 2054221614063184107682218077003539824552559296000 and 457936×10⁹¹⁷.

357686312646216567629137 ~= 3.58×10²³

Starting with a prime, you can often add a digit to the beginning to make another prime. For example: 7 -> 17 -> 317 -> 6317 -> 26317, etc. 357686312646216567629137 is the largest number in base 10 that you can get this way. If 0 is allowed as a digit, then the task of finding the largest has no well-defined solution. (There are many, arbitrarily large, primes similar to 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003 = 10¹⁰¹+3). See also 33333331 and 73939133.

602214179000000000000000 = 6.02214179(30)×10²³

The current most-accurate form of the "Avogadro constant" that most of us learned in chemistry class as "6.02×10²³". This is the number of carbon atoms in 12 grams of pure carbon, unpound and in ground state. In general it is very close to the number of atoms in N grams of an element with "atomic weight" N. (The term atomic weight has since been replaced by "relative atomic mass".) For different substances the correspondence varies slightly, primarily because of binding energy in the nucleus and mass of the electron

A lot of people identify the Avagadro number as the largest number they have had to use, or in many cases, actually memorize. The (30) at the end of "6.02214179(30)" represents the standard uncertainty (error) in the value: It is a more concise way to say "6.02214179 plus or minus 0.00000030".

3608528850368400786036725 ~= 3.6×10²⁴

This number has the property that for any N, its first N digits are divisible by N (for example, 3608 is divisible by 4, 36085 is divisible by 5, etc.) There is no larger number with this property. See also 73939133.

1000000000000000000000000000 = 10²⁷

(Approximately) the power output of the Sun in watts.

10²⁷ is one octillion. Walt Whitman used octillions, as well as several other large number names including decillions, sextillions, quintillions, quadrillions, trillions, ten billion and ten thousand, in his poetry. See also 10¹¹.

5524751496156892842531225600 ~= 5.5×10²⁷

The number of 9×9 Latin Squares. See also 6670903752021072936960.

5.4×10²⁷

Approximate mass of the Earth, in grams. See also 1.988435×10³³.

7000000000000000000000000000 = 7×10²⁷

(Approximately) the number of atoms in a 70-kilogram (154-pound) human being. To calculate your number of atoms, multiply your weight in kilograms by 10²⁶.

10888869450418352160768000000 = 27! ≅ 1.0889×10²⁸

Just another gratuitous 27 reference. See also 10^{3.0056206947796095239×1029}.

5×10³⁰ = 5 nonillion

An estimate of the number of prokaryotes (tiny organisms without a nucleus, including bacteria) on the Earth³. Produced by William Whitman and colleagues at the university of Georgia, the estimate includes 26×10²⁸ organisms within the top 8 meters of the ground, 12×10²⁸ in the water and oceans, about 80×10²⁸ on land but below the 8-meter point, and 355×10²⁸ in the ocean floor. The same scientists measured the mutation rate and estimated that every 20 minutes, somewhere on Earth a new species of bacteria comes into existence. Since prokaryotes comprise the vast majority of all living cells⁶, 5×10³⁰ is also the total population of living things on the planet.

See also 10¹⁴.

19252884016114523644357039386451 = 1205967 × primorial(61) + 19111438098711663697781258214361 ≅ 1.9252884...×10³¹

The smallest known set of 7 consecutive primes that are spaced an even distance apart starts with this number. It was discovered by Laurent Fousse & Paul Zimmermann in 2004²¹. It is probably not the smallest; these types of primes are much more common at larger sizes, and therefore an exhaustive search for the first one takes longer than a coordinated search for a larger one. It is suspected that the smallest is about 22 or 23 digits long. See also 47, 251, 9843019 and 121174811.

91409924241424243424241924242500 = 1¹⁰ + 2¹⁰ + 3¹⁰ + 4¹⁰ + ... + 999¹⁰ + 1000¹⁰ ≅ 9.1409924...×10³¹

The sum of the 10^th powers of the numbers from 1 to 1000. Notice the pattern in the digits — sort of regular, yet sort of irregular.

Around the year 1690, Jacques Bernoulli was studying the task (which was common at the time) of computing sums of powers. In the process he discovered the Bernoulli numbers, and used his new formulas to quickly calculate this sum as a demonstration of the importance of his discovery.

Bernoulli Numbers

The Bernoulli numbers appear as coefficients in infinite series for many things including the sums of powers, combinatorics, the Gamma function, etc. The first few Bernoulli numbers are: B₀=1, B₁ = ^-1/₂, B₂=¹/₆, B₃=0, B₄=^-1/₃₀, B₅=0, B₆=¹/₄₂, B₇=0, B₈=^-1/₃₀, B₉=0, B₁₀=⁵/₆₆, B₁₁=0, B₁₂=^-691/₂₇₃₀, B₁₃=0, B₁₄=⁷/₆, B₁₅=0, B₁₆=^-3617/₅₁₀, B₁₇=0, B₁₈=⁴³⁸⁶⁷/₇₉₈, B₁₉=0, B₂₀=^-174611/₃₃₀, B₂₁=0, B₂₂=⁸⁵⁴⁵¹³/₁₃₈, B₂₃=0, B₂₄=^-236364091/₂₇₃₀, ... Notice that the odd Bernoulli numbers starting with B₃ are all zero, and the even ones starting with B₂ alternate in sign. Also, after a point the size of the numerator starts to grow quickly. To compute the Bernoulli numbers, most mathematicians use the "generating function" method:

f(x) = x / (e^x-1) = SUM_n=0...inf[B_n xⁿ / n!]

The way you determine the B_n from this is by finding the value of the n^th derivative of the function at 0, which is equivalent to calculating the coefficients of the Taylor series for the function:

f(x) = f(0) + f'(0) x / 1! + f''(0) x² / 2! + f'''(0) x³ / 3! + ...

Since f(x) and each of its derivatives are undefined for x=0, you actually have to use the limit as x approaches 0. Or, you can use a symbolic math program — for example, in maxima type taylor(x/(%e^x-1),x,0,10); then multiply the n^th coefficient by n!

If you don't want to go to all this trouble the Bernoulli numbers can also be calculated by a simple iterative algorithm:

B₀ = 1
B_n = - n! SUM_k=0..(n-1) [B_k/(k! (n+1-k)!)]

The sums for the first few B_n's expand out like so:

B₁ = - 1! ( B₀/(0!×2!) )
B₂ = - 2! ( B₀/(0!×3!) + B₁/(1!×2!) )
B₃ = - 3! ( B₀/(0!×4!) + B₁/(1!×3!) + B₂/(2!×2!) )
(etc.)

The factorials get to be rather big pretty quickly. You can avoid such big numbers by using a different, less elegant iterative algorithm, illustrated here:

B₀ = 1
B₁ = - B₀ / 2
B₂ = (-3 B₁ - B₀) / 3
B₃ = (6 B₂ + 4 B₁ + B₀) / 4
B₄ = (10 B₃ - 10 B₂ - 5 B₁ - B₀) / 5
B₅ = (15 B₄ - 20 B₃ + 15 B₂ + 6 B₁ + B₀) / 6
B₆ = (21 B₅ - 35 B₄ + 35 B₃ - 21 B₂ - 7 B₁ - B₀) / 7
(etc.)

The B₃ and following lines all fit a pattern: the coefficients of the B_n come from Pascal's triangle, and the signs alternate, except for the sign of B₁ which should be inverted (to make it have the same sign as B₂ and B₀).

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