| Notable Properties of Specific Numbers |
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This is 29 primorial, 2×3×5×7×11×13×17×19×23×29 and has a really easy-to-remember digit pattern: 646 969 323 0. The pattern results from the properties of 1001=7×11×13 and 2001=3×667=3×23×29, which multiplied together give 2003001, and 323=17×19.
At the time of this writing, the appriximate estimated exact population of the world, from this site.
See also 1×1014.
Alternate answer to the "first prime number in alphabetical order" question (see 8018018851).
This is the first prime number in alphabetical order in the English language: "eight billion eighteen million eighteen thousand eight hundred and fifty-one". It was found by Donald Knuth. All other numbers that occur earlier in alphabetical order (like 8 and 8018018881) are composite.31
(Neil Copeland has suggested that 8000000081 comes earlier, based on the spelling "eight billion and eighty-one". Presumably Knuth leaves out the and.32)
The sixth perfect number. The even perfect numbers (it is not known if there are any odd perfect numbers) can all be expressed in the forms:
2P-1 (2P - 1)
2N (2N+1 - 1) where P is a prime and N = P+1. In this
example, P is 17. Also, for the number to be perfect, 2P-1 must
be prime, and is called a Mersenne prime.
See here for a complete list of known perfect
numbers.
See also 4.4823309×1014471464.
Years in the Hindu "Day of Brahma" (see 622080000000000).
Ten billion. See 22026.465794806 and 101010.
The largest number that can be formed from the digits 1, 2 and 3 using the ordinary functions addition, multiplication and/or exponents. It slightly edges out 231=2147483648 because log(3)/log(2) is greater than 3/2. The next number in this sequence is 101.0979465941272×1019.
This is the length of the day in paramanus, a small unit of time in
Hindu mythology. There are many different Hindu time systems that
contradict each other; this is one of the more irregular ones.
| 1 paramanu = 3.29 usec | |||
| 2 paramanus | = 1 anu | 1 anu = 6.58 usec | |
| 3 anus | = 1 thrasarenu | 1 thrasarenu = 19.8 usec | |
| 3 thrasarenus | = 1 thrudi | 1 thrudi = 59.3 usec | |
| 100 thrudis | = 1 talpara | 1 talpara = 5.93 msec | |
| 30 talparas | = 1 nimesha | 1 nimesha = 178 msec | |
| 3 nimeshas | = 1 kshana | 1 kshana = 533 msec | |
| 6 kshanas | = 1 kashta | 1 kashta = 3.2 sec | |
| 15 kashtas | = 1 laghu | 1 laghu = 48 sec | |
| 15 laghus | = 1 khadi | 1 khadi = 12 min | |
| 2 khadis | = 1 muhoorta | 1 muhoorta = 24 min | |
| 7.5 muhoortas | = 1 yama | 1 yama = 3 hours | |
| 8 yamas | = 1 ahoratra | 1 ahoratra = 1 day
|
This number's appearance in a 9th-century document from India helps prove that our place-value number system originated there. The description reads "ekâdishadantâni kramena hînâni", translated "beginning with one [which then grows] until it reaches six then decreases in reverse order". We know that the number being described is 12345654321 because it follows a description of the calculation of 111111×111111.
Current estimate of the age of the universe in years. This number comes out of calculations from data gathered by the WMAP mission, which observed the cosmic background radiation very accurately. It also assumes the Lambda-CDM model of the universe, which is the simplest model consistent with everything that has been observed so far (as of early 2008). This model also requires the existence of dark energy and dark matter, and several predictions of cosmic inflation including a flat universe far larger than the observable universe. If all this is true, then the figure (13.73 billion years) is known to be accurate to within 0.9 percent. See also 1040 and 8.02×1060.
23295638016 = 28563 ≅ 2.329×1010
This is the smallest cube that can be expressed as a sum of a sequence of consecutive cubes in more than one way: 28563 = 2133 + 2143 + ... + 5553 = 2733 + 2743 + ... + 5603. See also 216 and 8000.
61917364224 = 1445 ≅ 6.191×1010
This is the smallest 5th power that is also the sum of four distinct 5th powers: 1445 = 275+845+1105+1335. All bigger solutions up at least as high as 24485 are just multiples of this one. Because 144 is a square, this also happens to be the smallest 10th power that is the sum of four distinct 5th powers. There are plenty of other numbers that set "smallest sum of powers" records; for a long time, searching for such sums was very popular because of their similarity to the Fermat's Last Theorem problem.
What's particularly interesting about this number is the serendipitous way in which it was discovered. In 1966, two mathematicians programmed a computer to find solutions involving a sum of five 5th powers. They had no idea that there might be a solution with only four 5th powers, and in fact Euler had conjectured that there were none. But, due to an error in programming, the computer treated 0 as a valid number, and one of the solutions they found included 05 as one of the five 5th powers. As a result, they were pleasantly surprised to find that they had disproven a conjecture of Euler purely by accident!
Current estimate of the distance to the edge of the visible universe, in light years (in Planck units it is a much more impressive 8.02×1060). The matter that produced the cosmic background radiation, emitted 13.7 billion years ago, has subsequently traveled away from us (while forming into galaxies) and is now about 46.5 billion light years from us. This is a "comoving distance", a cosmological term which roughly describes a distance one would get if it were possible to measure "instantly" without the effects of relativity. There seems to be a paradox — moving at the speed of light, such matter could only have gotten as far as 27.4 billion light years.37
Number of particles (protons, neutrons, and electrons) in a bacterium, based on the figure 10-12 for the mass of a bacterium1. See also 713580, 1014 and 5.1843×1022652507173.
An estimate of the total number of people that have ever been born (assuming Darwin and evolution). The number depends highly on the decision on when pre-human apes begin to count as "people". This estimate draws the line at about 55000 BC. See this page for an explanation of the formulas used to compute the number. See also 5×1030.
100000000000 = 100 billion = 1011
This number is the na-yu-ta (なゆた) of the Lotus Sutra as it appears in e.g. Tendai schools of Japanese Buddhism. Is is descended from the Sanskrit niyuta as used in e.g. the Lalitavistara sutra. There are many other powers of 10 that have special names in Sanskrit; see also 1059.
1011 is one of the largest numbers explicitly named in a published music lyric. It is heard in Message in a Bottle by The Police, which has the lines "100 billion bottles / washed up on the shore". Sting is from England, but the song was written well after 1974, so I suppose he would have been thinking of the Chuquet billion, 109, when writing the lyric. See also 1012, 1027 and 101010.
An astronomical unit (AU) in meters, which is approximately the distance from the Earth to the Sun. It is a common unit of measurement in astronomy; for example, the distances of extrasolar planets from their parent stars are usually given in AU.
For scientific purposes it is defined as the distance a particle would need to be to orbit the Sun in one Gaussian year. This is not quite the same as the Earth's semi-major axis mainly because of the difference between the Gaussian and sidereal years.
Because of the agreement linking the Imperial and Metric length units, the astronomical unit is precisely this value regardless of changes in the Earth's orbit or the accuracy of our measurements thereof. However, it does remain dependent on the solar mass and the gravitational constant.
See also 9.46×1015.
Five hundred billion, the largest value I have seen on paper currency: the 500000000000 Dinar note which was issued in Yugoslavia as part of the 1993 series during their civil war and hyper inflation. Here is an image. This hyperinflation was much worse than the German 1923 episode but not as bad as Hungary's experience during World War II. However, the Yugoslavs probably have the record for the largest number on a piece of currency (the Hungarian "100 million B-pengo" represents 1 000 000 000 000 000 000 000 = 1021 Pengo, but the number on the note was "egy-milliard", 1 000 000 000, because the currency had been reissued in units of "B-Pengo" with one B-Pengo equal to 1012 of the older Pengos. If you want to call it the highest-denomination note, then you'd also have to count the Yugoslav 1994 10000000 Dinar note which is equivalent to 1027 pre-1990 Dinars). The Yugoslav currency was re-issued four times, in 1990, 1992, fall 1993 and January 1994, with multiplier fctors of 10000, 10, 106 and 109 respectively, thus the 1994 Dinar was worth 100000000000000000000 (1020) pre-1990 Dinars. (For more about record-setting currency, visit this site.)
A trillion in the original Chuquet system, as is used in most of the English-speaking world. In the United Kingdom before 1974, a "trillion" was 1018.
This is (as far as I am aware) the largest number explicitly named in any published music lyric. It is heard in Unacceptable by Bad Religion, which talks about chemical pollution, contains the lines "It's one part per trillion... unacceptable / One part per billion... unacceptable / One part per million... unacceptable". See also 101010.
This is the length of the day in thruti, a small unit of time in
Hindu mythology. There are many different Hindu time systems that
contradict each other; this one is notable for involving a fairly
orderly set of powers of 60, and also seems to involve the largest
total ratio between smallest and largest unit.
| 1 thruti = 18.5 nsec | |||
| 100 thruti | = 1 thatpara | 1 thatpara = 1.85 usec | |
| 60 thatparas | = 1 para | 1 para = 111 usec | |
| 60 paras | = 1 vilipatha | 1 vilipatha = 6.67 msec | |
| 60 vilipathas | = 1 liptha | 1 liptha = 0.4 sec | |
| 60 lipthas | = 1 vighati | 1 vighati = 24 sec | |
| 60 vighatis | = 1 ghati | 1 ghati = 24 min | |
| 60 ghatis | = 1 ahoratri | 1 ahorati = 1 day | |
| 60 ahoratri | ≅ 1 ritu (season) | ||
| 60 seasons | = 1 decade |
|
The system can be extended, as I have shown, to longer periods of time. The 60-day "season", (one sixth of the solar year, actually closer to 61 days), is used by many Hindus; the seasons are spring, summer, monsoon, autumn, winter, and dewy19. There is another Hindu time division system that divides the day into powers of 30. See also 405000, 26244000000.
This is 327, which can also be expressed 333, 3③3 or 3④2. The latter two forms use the hyper4 and hyper5 operators.
A light year in kilometers.
Estimates of the number of cells in a typical human's body range from 1013 to 1014.
1014 is also an estimate of the number of bacteria living on or in a human body.2 Another source3 gives the value 6.5×1013. Most of these are either in the skin or somewhere inside the digestive tract (mouth, intestine etc.) and provide essential support to digestion and other vital functions. See also 6788516573, 75250000000, 1019 and 5×1030.
1014 is also an estimate of the computing rate (in floating-point operations per second) needed to simulate a human brain. See 1019.
This is the product of two primes, 3536123 × 128541727, and also the sum of these two plus all the primes in between: 3536123 + 3536129 + 3536131 + ... + 128541719 + 128541727. Thus it has the "39 property"; it is the 5th number with this property after 10, 39, 155 and 371.
There is also a reasonably good chance that it is the largest, but it's hard to tell. As N gets larger, the odds of this property happening fall off quite quickly. If there is another, it is greater than 3.63 × 1018.
This is an example of a phenomenon that is a counterpart to Guy's Strong Law of Small Numbers (see 91), and unfortunately not called the "Strong Law of Large Numbers" (which is an unrelated and much more formal law of statistics).
If two properties are unrelated, and if the likelihood of occurrence of both properties for any one integer N falls "quickly enough" then the set of such numbers is probably small and finite. (If it were not small and finite, there would be a relation between the two properties.)
Without being rigorous and getting into heavy number theory, you can often make a pretty good guess as to whether a set of numbers is finite or not by a brute-force search. If the numbers found by the brute-force search suggest a probability of less than 1/N for any given integer being in the set, then the set is probably finite. If the probability is 1/N or more, the set is probably infinite.
To give a simple example: consider a hypothetical "search" for powers of 2. The odds of any integer N being a power of 2 can be stated as log2(N)/N, because there are log2(N) powers of 2 spread out over the first N integers. log2(N)/N is bigger than 1/N, so that implies that there are an infinite number of powers of 2. A computer search would "verify" this by showing that for the first N integers there are approximately log2(N) powers of 2.
When the set of numbers is even more rarely distributed, the method isn't really useful. For example, consider integers of the form 22n. For each N the probability is log2(log2(N))/N, only slightly more than 1/N. The numbers are so spread out that a brute-force search doesn't give enough data to make a clear decision. It could spend less than a second finding the first 5, and years before finding the 6th. This is why we can't be too certain that 454539357304421 is the highest number with the "39 property".
An example of a set of numbers that is right "on the edge" of decidability by this method is the Mersenne primes. These are primes (probability roughly 1/ln(N)) that are 1 less than a prime power of 2 (probability log2(N) ln(log2(N))/N). The combined probability is ln(2) ln(log2(N))/N, or ≅ 5 ln(2) ln(log2(N))/N if you account for the fact that Mersenne numbers are known to be relatively prime to all smaller Mersenne numbers. These odds fall only a tiny bit less fast than 1/N — too close to call from a brute-force search alone. In fact, the formula K ln(2) ln(log2(N))/N agrees very closely with the number of known Mersenne primes, although K is somewhat less than 5. The search for more Mersenne primes is ongoing, and much research has yielded neither proof nor disproof that they are infinite in number.
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.
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.
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".
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.
According to Richard Crandall67, the "longest calculation ever performed" to get a yes-or-no answer, is 1016 fundamental operations, to prove that the 22nd 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.
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... = eπ√163 ≅ 2.625×1017
An example of a "coincidence" that isn't. The value of eπ√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:
eπ√67=147197952743.999998662..., approximately equal to
5280^3 +744 =(25×3×5×11)3+744
eπ√43=884736743.9997774..., approximately equal to
960^3 +744 =(26×3×5)3+744
eπ√19=885479.7776..., approximately equal to
96^3 +744 =(25×3)3+744
eπ√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:
eπ√58=24591257751.9999998222...,
eπ√37=199148647.99997804....,
eπ√22=2508951.998257...,
There are far more than the statistically-expected number of occurrences of eπ√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=eπ√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.
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