Notable Properties of Specific Numbers
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According to my classical sequence generator, 1011 is the next number after my "favorite" numbers 3, 7, 27 and 143. The formula it finds is: A0 = -1; AN+1 = (2N-1)(AN+1) + 3 (sequence MCS8041809, with its own whimsical page here). Along with its successor 9111, the terms in the sequence share common factors and other properties with 3, 7, 27 and 143. This serves as an example of how easy it is to find a sequence formula to match an arbitrary set of numbers. See also 695, 715, and 1011.
The number of ways to form a set of yes-or-no questions that can be used in a "20 questions"-like game where the questioner knows that the item to be guessed is one of a specific pre-defined set of 8 items. For 8 items you always need at least 3 questions and might need as many as 7. This problem is the subject of Sloane's integer sequence A5646, and I have written a thorough discussion of it here.
This is 210, and it's pretty close to 1000=103. As a result, and because of the long-established habit of grouping digits in threes (see 1000), the computer industry has adopted the international prefixes kilo, mega, etc. to refer to quantities that are actually powers of 1024 almost as if they were actually powers of 1000. This sometimes causes practical problems and confusion which can be avoided by using the official prefixes kibi, mebi, gibi, etc. (There is a table here and more about SI prefixes here.)
If for some reason you decide to learn the powers of 2, the closeness of 1024 and 1000 makes it a little easier. It's also handy that 1024 is the 10th power of 2 since 10 is the base of our number system, so (for example) 27, 217, 227, 237 and so on all start with roughly the same digits. Here is a table of powers of 2:
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Given the importance of the powers of 2 for such things as the size of the memory chips in your computer, it's actually pretty cool that we have this coincidental closeness to a power of 10, and a popular power of 10 to boot. It didn't have to be that way. The only other powers of numbers that come anywhere close to a power of 10 are things like 223=10648 and 3162=99856, and those aren't too useful because there isn't much in real life that involves powers of 22 or 316.
1080 = 27×40 = 15×72 = 23×33×5 = 20×19×18/1×2×3
A fairly highly-composite number (but not a record-setter) which appears as a unit of division in the Talmudic Hebrew time system. See also 108.
See 8712.
Appears in Star Wars as a reference to George Lucas' earlier movie THX 1138. See also 2187.
This number has 36 distinct divisors: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, and 1260. No smaller number has so many divisors, so 1260 is a divisibility record-setter. The fairly popular numbers 8, 24 and 40 are missing from this list; 1260 is the largest such record-setter not divisible by 8.
Numbers with lots of divisors were popular in ancient civilizations; well-known examples include 12, 24, 60 and 360. 1260 is not as famous but it does appear in the Bible, both explicitly in Rev 12:6 and implicitly with the phrase "a time, and times, and half a time"12 in Rev 12:14.
"A time, and times, and half a time" is generally taken to be a reference to a period of 31/2 years12,13, and an allusion to similar phrases in Dan 7:25 and Dan 12:711. The "years" in question are probably the Babylonian "lunar years" of 360 days, so 31/2 years is 1260 days — the same time period referred to explicitly in Rev 12:6, and described as "42 months" (42×30 days) in Rev 11:2 and Rev 13:5. All of these are meant to refer to a "prophetic", not literal, period of time in which "1 day" in the text represents one year in real life (this is kind of like the Hindu manvantara, see 4320000000). In other words, the period being referred to is 1260 years, or perhaps 1260 Babylonian lunar years which would be 453600 days.
Round numbers with more divisors were more likely to show up in ancient writings, partly because of the difficulty of manipulating "odd" numbers accurately. 1260 might have been more appealing to the ancients because 1260 days is exactly 180 weeks (180 being another record-setter), or perhaps because 1260 is 12×100+60.
There are many other numbers involved in apocalyptic predictions, such as 1290, 1335 and 2300 (intervals of years and of days mentioned in the book of Daniel); 945000=360×(1290+1335); etc.
Mentioned in the Old Testament apocalyptic book Daniel (see 1260).
The sum of the first 8 cubes, and also 362 and 64. See also 216.
1331 = 113, a fact that holds in all bases higher than 3 (where you get "2101"). See 121.
1331 resembles a row of Pascal's triangle; see 14641.
Mentioned in the Old Testament apocalyptic book Daniel (see 1260).
An alternate spelling of leet; see 31337.
1353 = 13+14+15+...+52+53, a member of sequence A186074 and its subset {15, 1353, 133533, 13335333, ...} all of which have the same property. See 429 for more. (Contributed by Matt Goers)
The number of ways to pick 5 numbers from 1 to 25 (with no two the same) and have them add up to 65.
1460 is 365×4, and therefore the number of years that have to pass (in a Julian calendar) to accumulate an entire years' worth of leap days. This was known to the Egyptians as the Sothic cycle, the number of 365-day calendar years that have to pass for the calendar to once again agree with the seasons.
See also 1508.0833.
The number of days in a "quadrennium" or "Olympiad" of 4 Julian years.
1508.0833 = 365.242189670 / 0.242189670
The number of years in the Sothic cycle, which is the amount of time it takes for dates in a constant 365-day calendar to drift all the way around the seasons and arrive back where they started. The drift was noticed by the Egyptians shortly after they had established their 365-day calendar, at a time when the rising of Orion at sunset coincided with the flooding of the Nile. They noticed the drift but stuck to their calendar (being close enough to the equator that the seasons didn't affect the length of their day too much). The 365-day Egyptian calendar continued in unbroken use for two (or perhaps three) entire cycles. After a cycle had been completed, by looking back at the written record of Pharaohs' reigns, they could calculate the length of the cycle.
On the advice of the Alexandrian astronomer Solsigenes, the approximation 1460 = 365×4 was used by Julius Caesar for the Julian calendar. The Egyptian calendar (with its original month names and 5-day intercalary month coinciding with the beginning of the Nile flood period) remained in use and was also modified to a 365.25-day cycle; it is still used to this day (as the Coptic Orthodox Church's liturgical calendar).
In the year 1514, German artist Albrecht Durer created one of his better-known works, Melencolia I, a still-life containing many symbols of alchemy including a 4×4 associative magic square containing the numbers 15 and 14 in adjacent positions. The full 4×4 square appearing in Melencolia is:
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Each row (such as 16+3+2+13), column and diagonal adds up to 34. It has to be 34 because the sum of all 16 of the numbers must be the sum of four rows, and also the sum of all 16 numbers, thus we have 4M=16×17/2 where M is the magic sum, thus M=34.
Also, the four numbers in any one quadrant (for example, the upper-left quadrant, 16+3+5+10) add to 34. And because this magic square is "associative", there are a lot of other sets of 4 squares that add to 34, such as the four corners, the four in the center, and other symmetrical patterns such as 3+2+15+14, 16+6+11+1, etc. Any pattern shaped like one of the following works (a total of 28 patterns):
10 4 11 4 12 1 14 2 15 2 20 2 X X . . X . X . X . . X X . . . X . . . . X X . . . . . . . . . . . . . . X . . . . X . . . . . . . . . . . . . . . . . . . X . . X . . . . . . . . X X . X . X X . . X . . . X . . . X . X X . 22 2 23 4 24 4 25 2 60 1 . X . . . X . . . X . . . X . . . . . . X . . . . X . . . . X . . . . X . X X . . . . X . . X . . X . . X . . . . X X . . . X . . . X . . . X . . . X . . . . .
The length of a mile in meters. This is exact, since the inch is defined as precisely 2.54 cm. See also 1.609344, 63360 and 1609344.
Number of years from the Creation to the Flood in the Hebrew tradition (and Judeo-Christian Bible). See 86400 for more details.
In 1974 a radio message was sent from the Arecibo observatory towards star cluster M13 (a group of stars about 25,100 light years away) containing 1679 bits of data modulated by frequency shift keying. The number 1679 was chosen because it is a semiprime — a receiver of the message would presumably notice this, then try to arrange the data into a 23×73 or 73×23 rectangle to look for a pattern.
This is 8×7×6×5 = 8! / 4!. Numbers of the form 2N!/N! are called quadruple factorials. The quadruple factorials are: 1, 2, 12, 120, 1680, 30240, 665280, 17297280, ... (Sloane's A1813).
Confusing, but there is another definition of "quaruple factorial" that is more like the "double factorials", in which you form a product by starting with some integer N and subtracting 4 each time: 1680 = 14×10×6×2. Using this definition, all of the 2N!/N! numbers are included, plus three intermediate values between each one: 1, 1, 2, 3, 4, 5, 12, 21, 32, 45, 120, 231, 384, 585, 1680, 3465, 6144, 9945, 30240, 65835, 122880, ... (Sloane's A7662). See also 105.
(I happen to think that this name "quaruple factorial" is even worse than "double factorial", but that's just me.)
1680 is also a highly composite number.
The product of three consecutive integers (a 3-d oblong number), and also one of the central numbers in Pascal's Triangle, namely the 7th term in row 13. This coincidence happens because 7! = 5040 = 10×9×8×7 (and therefore, 13!/(7!×6!) = 13!/10!); see 3628800 for more on this.
This is 123, the number of cubic inches in a cubic foot. It is sometimes called a great gross53. See also 1729.
One of the numbers Ramanujan made famous. As the story goes, Hardy commented to Ramanujan that he had come over in taxicab number 1729 and that the number had no particular significance that he (Hardy) knew of. Ramanujan replied that 1729 did indeed have a special property: it is the smallest number that can be expressed as the sum of two cubes in two different ways: 1729 = 123+13 = 103+93. It is thus also a near-miss to Fermat's Last Theorem. See also 50, 65 and 635318657.
This feat seems extraordinary, and most write it off to the fact that Ramanujan had a sort of savant calculation ability. However, it is a little easier to see how this particular feat can be done by considering the following:
First of all, once a modest-sized list of cubes has been memorized (something that many folks with a passion for numbers do, see 7776 and the Feynman anecdote below), it is easy to recognize 1729 as the combination of 103=1000 and 93=729, and it also has a clear resemblance to 123=1728.
It then remains to determine if 1729 is the lowest such number. There are a lot of sums to check — searching for a solution smaller than 1729, there are almost a hundred ways to add two cubes and get a total that is less than 1729. The sums are: 1+1=2, 1+8=9, 8+8=16, 1+27=28, 8+27=35, 27+27=54, 1+64=65, ... (Sloane's A3325). Somehow you have to mentally look through the "list" to find if any of these occur twice.
An important insight, and the type of thing that Ramanujan is sure to have noticed, is that the mapping N→N3 is invariant modulo 6. In other words, if a given number is of the form 6N+K for some N and K, then its cube will be of the form 6M+K, with the same value of K (Another way to express this is that N3-N is always divisible by 6). Here are the first few cubes expressed as 6M+K with the K part in bold: 13=0×6+1, 23=1×6+2, 33=4×6+3, 43=10×6+4, 53=20×6+5, (6+0)3=36×6+0, (6+1)3=57×6+1, (6+2)3=85×6+2, and so on.
Here are the cubes up to 123 classified with letters a through f for the 6 different values of K:
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The sum 729+1000 is a c plus a d, with a sum of type 6N+1. To get another sum of type 6N+1, there are only two types of changes that can be made: you can either move one number down to a b and the other up to an e, or you can move them both down three spaces to make it an f plus an a. Either of these types of changes can be repeated to get another candidate pair. Neither of these changes alone can possibly result in the same sum: the first always results in a larger sum, and second always results in a smaller sum — so you have to perform at least two such actions to have any hope of getting a match. Also, we can ignore sums that come from moving the pair up three spaces, because we can assume that our starting pair is the one with the higher "center" (otherwise, we'd be finding each solution twice, and there is no need to do that).
So, starting with 729+729=1458, we'll find another pair with the same sum but where the center of the pair is lower. Moving down to 216+216, then trying 125+343, 64+512, 27+729, 8+1000 and 1+1331, all are too small. Starting with 27+27, it's clear all possibilities will be too small so we're done with 729+729.
The next starting pair is a wider pair with the same center: 512+1000=1512. Moving down to 125+343, we get the same sequence again ending with 1+1331, still all too small. This time, notice that the "target" 1512 is even bigger than our previous "target" of 1458, so we didn't even have a chance of making the target. Thus, any wider pairs on the same center (which includes just one, 343+1331=1674) will fail in the same way, and the pair after that, 216+1728, is bigger than the known solution 1729 so we're done checking pairs centered at 729.
Now we go to 512+729=1241. Moving down to 125+216, and checking each pair up to 1+1000=1001, they are all too small. In the same way just described, the pairs 343+1000=1343, 216+1331=1547 cannot possibly produce a match and 125+1728 is too big.
Now we go to 512+512=1024, and the same thing happens (the closest we get is 1+729=730)
Then we check 343+512=855, now we get no closer than 1+512=513. Beyond this point, it's easy to see that if our first pair is A+B, the second pair never gets any higher than 1+B. For example, starting with 343+343=686, the closest match is 1+343.
Another lovely anecdote about 1729 involves Richard Feynman, the physicist from Cal Tech. Feynman was in a Japanese restaurant and ended up in a sort of mind-vs-machine contest with an expert abacus operator. After losing to the abacist in addition (handily) and multiplication (a closer race) then coming up dead even in long division, he was challenged to extract the cube root of "any old number", and the number they were given, intentionally chosen at random, turned out to be 1729.03. Feynman remembered that 1728 is 123, so in his head he did the following (which can be derived from the derivative for xk, or a simple inversion of the binomial expansion for (a+b)3):
(1728+1)1/3 = 12 (1+1/1728)1/3 = 12 (1 + (1/3)(1/1728) + ...)
≅ 12 + 1/432
He then began performing the long division 1/432 in his head, and got as far as "12.002.." before being proclaimed the winner.43 Since 12+1/432 = 12..0023148148... and 1729.031/3 = 12.0023837856..., he could have given one more digit of 1/432 and still been correct.
1729 is also the third Carmichael number. Its factors are 7×13×19; J. Chernick proved that any number of the form (6n+1)×(12n+1)×(18n+1) is a Carmichael number provided that all three are prime. The "Chernick numbers" are: 1729, 294409=37×73×109, 56052361, 118901521, 172947529, ... (Sloane's sequence A33502).
The mass ratio between the proton and electron. It sometimes attracts attention similar to that given to the fine-structure constant.
The mass ratio between the neutron and electron. It sometimes attracts attention similar to that given to the fine-structure constant.
The number of meters in a nautical mile. This unit was originally defined as the length of one minute of latitude along a meridian (or, more approximately, any great circle) on the Earth. This makes the circumference of the Earth 360×60=21600 nautical miles long. This was done for a utilitarian reason — you can take a distance on a chart, measure it against the latitude gridlines on the edge of the map, and that tells you how many nautical miles long it is. The approximation varies with location because the Earth is not a perfect sphere; most of the variation is a gradual decrease in length from pole to equator. See also 5280 and 20003931.4585.
1951 is a prime number, and curiously the year in which the record for largest-known prime was broken for the first time in 75 years. The record was broken twice in that year — the first time by Ferrier using a mechanical desk calculator, then a second time by Miller and Wheeler using an electronic computer. 34
2001 = 3×23×29, three distinct primes which are distinct from the three primes that make up 1001. This causes the rather nice digit pattern of the primorial 6469693230=2003001×323×10.
Because the first year was year 1, and 1+2000=2001, 2001 is the 2000th anniversary of the year 1 (in whatever calendar you wish, most recently this happened in the Christian calendar). Although New Years revelers made a bigger deal about 2000, 2001 was considered the "real" millennium year by people who care about such things. (If you disagree, you are in good company. See here.)
A popular apocalyptic date (usually given as the date of the winter solstice, December 21st) because it is close to a hypothesized rollover date for a certain Mayan calendar (see 5126). That calendar was more likely designed to roll over in the year 4772 AD (on October 13th according to 80).
A Mersenne number that is not prime; see 496 and 8384512.
The largest known power of 2 whose digits are all even. Higher powers of 2 have been checked at least as far as 27725895275426. One does not need to check all of the digits because (for example) you can check just the last 20 digits and the odds are only 1 in 220 that all will be even.
See also 2.95×1020.
The product of 27=33 and 81=34, and containing the same four digits (see also 8127).
The house number of the childhood home of Martin Gardner, the longtime writer of the Mathematical Games column of Scientific American responsible for so many amateur mathematicians' introduction to popular mathematics. In one of his columns his character Dr. Matrix described many properties of 2187: it is the 297th lucky number; add its digits in reverse (7812) and get 2187+7812=9999; its digits can also be arranged to make 1728 and 8127, etc.
George Lucas was inspired by Arthur Lipsett's film mashup 21-87. "The force" comes from a line by Roman Kroitor in this movie. In a tribute reference, when Han Solo and Luke rescue princess Leia in Star Wars, they find her in cell 2187 (within cell block 1138).
2202 yojana is a distance that figures in a now-famous comment by Sayana a minister to King Bukka I of 14th-century India, in his commentary on the Rigveda. The full quote is "[it is] remembered that the sun traverses 2,202 yojanas in half a nimesha". A yojana is a unit of distance, whose definition varies throughout history and by context. It is agreed that a yojana is 4 kro'sha, but the definition of kro'sha can be either 1000 or 2000 dhanu. As a result, a yojana is either 4.5 or 9 miles (other sources say 5, "about 8 to 10", and 40). But if the figure of 9 miles is used, the speed of the sun is 39636 miles per nimesha. A nimesha is 1/405000 of a day, so converting to standard units, we have 299128 kilometers per second — very close to the speed of light. This is usually taken as being much more significant than mere coincidence would suggest, with the implication that Sayana was actually speaking of the speed of light, not of the Sun. However, it was common to estimate the speed of the Sun in its daily "orbit" in the old geocentric cosmology model, and some Hindu/Indian estimates are comparable. For example, in the Vayu Purana, chapter 50, the Sun is said to move 3150000 yojana in 1/30 of a day, or about 16000 kilometers per second. See also 405000
See also 299128000 and 309467700.0.
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Quick index: if you're looking for a specific number, start with whichever of these is closest: 0.065988... 1 1.618033... 3.141592... 4 12 16 21 24 29 39 46 52 64 68 89 107 137.03599... 158 231 256 365 616 714 1024 1729 4181 10080 45360 262144 1969920 73939133 4294967297 5×1011 1018 5.4×1027 1040 5.21...×1078 1.29...×10865 1040000 109152051 101036 101010100 — -- footnotes Also, check out my large numbers and integer sequences pages.
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