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757 is the smallest factor of 999999999999999999999999999=1027-1 that is not also a factor of a smaller string of 9's, and therefore 757 is the smallest number whose reciprocal has a 27-digit repeating decimal: 1/757=.00132100396301188903566710700132... This is part of a series: 1/3 has a 1-digit pattern, 1/27 has a 3-digit pattern, and 1/757 has a 27-digit pattern. The series keeps going up (because if 1/n has a d-digit pattern, n is always larger than d), but computing the next number in the series is hard because it is equivalent to factoring a large number like 10757-1. See also 7, 27 and 239.
Like 6, 12, 24, 48, 96, 192 and 384, 768 is a 3-smooth number of the form 3×2n. It is one of several such numbers to occur in personal computer display dimensions (the standard 1024x768 "XGA" mode); see also 192.
See 82944.
840 is a factors record-setter, and is the first that does not also appear as the number of factors of another factors record-setter. See also 12, 45360, 720720, 3603600, 245044800, 278914005382139703576000, 2054221614063184107682218077003539824552559296000 and 457936×10917.
841 = 292 is the sum of two consecutive squares: 202+212. This is the second smallest example; the smallest is 52 = 25 = 32+42. The series continues: 52, 292, 1692, 9852, 57412, 334612, 1950252, ... (Sloane's integer sequence A001653; my MCS13882118). This sequence consists of every other Pell number; Each of these is 6 times the previous one minus the one before that, for example, 169 = 6×29-5. See also 99 and 204.
952 is the sum of the cubes of its digits plus the product of its digits: 952 = 93+53+23+9×5×2. (Thanks to Cyril Soler for this tip)
1/998 = 0.001002004008016032064128256513026052104208416833667334669..., a repeating decimal in which the powers of 2 appear one after another (until they start to overlap and break the pattern)
0.001
0.000002
0.000000004
0.000000000008
0.000000000000016
0.000000000000000032
0.000000000000000000064
0.000000000000000000000128
0.000000000000000000000000256
0.000000000000000000000000000512
0.000000000000000000000000000001024
0.000000000000000000000000000000002048
. . .
0.001002004008016032064128256513026052...
The same thing happens to a lesser degree in the digits of 1/98, and to a greater degee in the digits of 1/9998, 1/99998, etc. The reason for the pattern is easy to see if you consider how long division is performed, or just notice that 1/998 = (0.998+0.002)/998 = 0.001 + 2 × (0.001 × 1/998). The same phenomenon is responsible for the powers of 3 in the digits of 1/997, and so on. A similar pattern involving the Fibonacci numbers appears in the reciprocal of 89.
The casting out nines principle works for 99, 999, 9999 and so on. For the same reason that "casting" gives us an easy way to test for divisibility by 3, it also allows us to test easily for divisibility by 11, since 11 is a factor of 99, and for divisibility by the factors of 999, which are 27 and 37. See also 101.
999 is an example of a "repdigit", a number consisting of the same digit repeated a number of times. Repdigits often come up in the study of numbers that have properties relating to the sum of their digits, repeating decimal fractions, etc. For example, many Kaprekar numbers are repdigits, or are closely related to them; repdigits and factors thereof also appear in the
Thousand is another number that, for most of us, temporarily holds the honor of being the biggest number we've heard of. Usually it replaces 100 in this role and is overtaken by 1000000.
When numbers get big, we tend to group the digits in 3's to make the numbers easier to read: 134,217,728 instead of 134217728. That causes people to pay a little more attention to groups of 3 digits than they otherwise would. For example, it's probably the main reason why I discovered this formula for 227. Sometimes it's almost as if we're working in base 1000.
The neighboring numbers 999 and 1001 also frequently play a role — if the phenomenon being investigated involves adding groups of 3 digits together, then it is related to divisibility by 999. If it involves a similarity between adjecent groups of 3 digits (such as 720720) it is related to divisibility by 1001. The two sometimes get inter-related when division is involved because 1/999 is 0.001001001001... and 1/1001 is 0.000999000999...; see also 999999.
1001 is a product of three consecutive primes: 1001 = 7×11×13. This sequence runs: 30, 105, 385, 1001, 2431, 4199, 7429, 12673, 20677, 33263, 47027, ... (Sloane's A046301). See also 77.
1001 is related in various ways to many numbers and number phenomena that have repeated sets of 3 digits, and some of these, like 720720, occur because 1001 is a product of three consecutive primes.
In One Thousand and One Nights, also known as 1001 Arabian Nights, concerns a very long period of time during which a series of stories and stories-within-stories are told by the narrator to avoid being killed by a king. There are many versions, including some that are actually organized into 1001 parts, all of suitable length for reading at bedtime and most of them ending in cliffhangers. 1001 nights is 143 weeks, nearly 3 years.
There is evidence of early versions of the Arabian Nights story collection, apparently an anthology of material originating in oral tradition, with far fewer than 1001 parts. 1001 was acquired and stuck because it represents "a lot" in a special way. The reader is probably familiar with similar uses of "101", "201", "501", and similar numbers in other contexts.
See also 999, 1000, 2001, 999999, 3603600, and this description of highly composite numbers.
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 A005646, 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. (see here for more about this.)
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:
| 20 = 1 | 210 = 1024 | 220 = 1048576 | |
| 21 = 2 | 211 = 2048 | 221 = 2097152 | |
| 22 = 4 | 212 = 4096 | 222 = 4194288 | |
| 23 = 8 | 213 = 8192 | 223 = 8388608 | |
| 24 = 16 | 214 = 16384 | 224 = 16777216 | |
| 25 = 32 | 215 = 32768 | 225 = 33554432 | |
| 26 = 64 | 216 = 65536 | 226 = 67108864 | |
| 27 = 128 | 217 = 131072 | 227 = 134217728 | |
| 28 = 256 | 218 = 262144 | 228 = 268435456 | |
| 29 = 512 | 219 = 524288 | etc. |
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 factors: 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 factors, so 1260 is a factorization record-setter. The fairly popular numbers 8, 24 and 40 are missing from this list; 1260 is the last factor record setter not divisible by 8.
Numbers with lots of factors 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" 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 factors 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 factors 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.
See 14641.
Mentioned in the Old Testament apocalyptic book Daniel (see 1260).
The number of ways to pick 5 numbers from 1 to 25 (with no two the same) and have them add up to 65.
1508.0833 = 365.242189670 / (365.242189670-365)
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.
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 Melancholia is:
| 16 | 3 | 2 | 13 |
| 5 | 10 | 11 | 8 |
| 9 | 6 | 7 | 12 |
| 4 | 15 | 14 | 1
|
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.
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 25,000 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. I think that name is even worse than "double factorial", but that's just me. See also 105.
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 A003325). 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:
1a 8b 27c 64d 125e 216f 343a 512b 729c 1000d
1331e 1728f
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 A033502).
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