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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 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
A product of 3 consecutive primes (see also 1001).
A permutation of the digits 1,2,3,4 that is also a multiple of 11 (see 132 for more about this; see also 163 and 101.0979×1019).
2520 = 23×32×5×7 = 5 × 7 × 8 × 9
2520 is the smallest number that is divisible by all the numbers from 1 to 10.
How do you find such a number? It doesn't need to be 1×2×3×4×5×6×7×8×9×10, there are many smaller answers. To consider a simpler example, 12 is the smallest number divisible by 1, 2, 3 and 4 but it is smaller than 1×2×3×4. There are different ways to find the answer, all of which amount essentially to looking at the prime factorizations of all the numbers and keeping the highest exponents of each prime that occur. The simplest way to describe the answer is as follows: Keep the highest power of each prime, and throw away all the rest. For example, in the numbers 1 through 10, the highest prime powers are 8=23, 9=32, 5=51 and 7=71. All the other numbers are composite (e.g. 6=2×3 and 10=2×5) or smaller powers of the primes already mentioned (2=21, 4=23 and 3=31).
This number is also related to 10 in another way: it is 10 times the central number (252) on row 10 of Pascal's Triangle. This is sort of related, because 252 = 10!/(5!×5!); see 5040 and 3628800.
2592 = 25×92, the only 4-digit number of the form ABCD = AB×CD66. (Thanks to Jim Cook for this one). See also 25920.
3003 occurs at 6 different places in Pascal's Triangle: in rows 14, 15 and 78, because 3003 = 14!/(6!×8!) = 15!/(5!×10!) = 78×77/2. This is related to the fact that 6=F3F4 and 15=F4F5 are two consecutive golden rectangle numbers. Numbers that occur at least 5 times in Pascal's Triangle include: 120, 210, 1540, 3003, 7140, 11628, 24310, 61218182743304701891431482520, ... (Sloane's A003015).
3003 is also a palindrome, and the 77th triangular number. It shares this property with 1, 3, 6, 55, 66, 171, 595, 666, and many others (Sloane's A003098). Note that the index (77) is itself a palindrome. Possibly because of the simple formula Tn = n(n+1)/2, there are also at least two cases with a repunit index: T1111=617716 and T111111=6172882716.
The solution to the "Monkey and Coconuts problem" (1926 Saturday Evening Post version), which according to Martin Gardner62,63 is "probably the most worked on and least often solved of all" Diophantine equations. Here is the problem (in my words):
Five sailors, stranded on a desert island, spend the day
gathering coconuts and then go to sleep, agreeing to divide them up
in the morning.
After a while, one sailor wakes up, concerned that he might
not get his fair share. He gives one coconut to the monkey to keep
him quiet, then divides the pile into five parts, and finds
that it divides evenly. He hides one fifth and puts the rest back
into one pile.
One after another, each of the other four sailors does the
same thing — wakes up, gives a coconut to the monkey, divides the
rest into five parts, which comes out evenly, hides one-fifth and
puts the remainder back into one pile.
In the morning, the men together divide the remaining
pile and find it divides equally into five parts. How many coconuts
were there in the beginning?
A little trial-and-error reveals that the answer is probably somewhat large. The general solution is nn-n+1 for odd n, and (2n-1)nn-n+1 for even n.64
See also 7.76×10202544.
3456 is twice 1728, and has a notable pattern of ascending digits. It also has the following cute relationship to 27: 27 × 27 = 3456.
This is e④π, where ④ is the lower-valued form of the hyper4 operator. (See here for discussion of the higher-valued hyper4 operator generalized to real numbers). See also 22.459157... and 4979.003621....
The number of seconds in an hour (60×60), the number of fingers in a cord (units of length), the number of shekels in a talent (units of weight), the number of years in the (Babylonian) long "saros", and the product of squares of the simplest Pythagorean triangle (32×42×52). All of these were significant to the Babylonians.
Although it is not a factor record-setter itself, 3600 is the square of the popular factor record-setter (and base of the Sumerian/Babylonian number system) 60.
The division of the hour into 60 minutes or 3600 seconds is the most familiar relic of the old Sumerian base-60 numbering system. It also survived in the divisions of angles into degrees, minutes (short for "minute divisions"), seconds, thirds14, and so on. The division of an hour into 3600 parts also happens to be convenient and useful. The hour had been long established as 1/12 part of the daylight period (to such an extent that, in many cultures, the length of the hour increased and decreased with the seasons!). The pace represented by 3600 beats per hour arises naturally because it is close to the frequency of a human heartbeat; most people feel thier heartbeat when resting quietly. Since hours and heartbeats were already pretty well established, it was useful to choose a number that was pretty close to the right ratio but also was arithmetically convenient to work with. 3600 was by far the best choice.
The first Fibonacci number with a prime subscript that is not itself prime: F19=4181=37×113.
4879 is a Kaprekar number under the most relaxed Kaprekar rules, in which the point of division can be in a place other than the number of digits in the original number. 4879^2 is 23804641, but instead of dividing it into 2380+4641, it is divided into 238+04641 = 4879. Even though the original number 4879 has four digits and its square has eight, the square is divided unevenly giving three digits to the "left half" and five digits to the "right half" 04681. Under the more strict Kaprekar rules, the division always happens in the place corresponding to the number of digits in the original number.
A square pyramidal number is a number in the sequence 1, 5, 14, 30, 55, ... that you get by adding consecutive squares starting with 1. The only square pyramidal number which is also a square is 702 = 4900.
See also 3.377×1038.
This is e④π, where ④ is the lower-valued form of the hyper4 operator. (See here for discussion of the higher-valued hyper4 operator generalized to real numbers). See also 23.140692... and 3581.875516....
5040 = 7! = 7×6×5×4×3×2×1 = 10×9×8×7. There are other numbers with a similar property (see 720 and 3628800)44.
5040 is also divisible by all the numbers from 1 to 10, and by 144 and several other useful numbers. Plato44 cited this as a reason for 5040 being the ideal number of citizens in a state.
5041 = 7!+1 = 712. This is the highest known case of a square which is one more than a factorial. The other cases are 25 = 4!+1 = 52 and 121 = 5!+1 = 112. There are several similar special properties of numbers (for examples, see 39, 89 and 51381) where the distribution falls off so quickly that it's difficult to see if there are only a finite number of numbers with the property. In this case for example, the odds of N! + 1 being a square are about 1 in √(N!), assuming there is no special relationship between the distribution of squares and factorials. Since the factorials grow very quickly, the infinite sum
SUM [ 1 / √(N!) ]
converges very quickly, and in fact it's a bit of a surprise that there are as many as three sulutions for N! + 1 = M2. The fact that there are three suggests that the distribution of factorials and squares might have a relationship.
An approximation of the number of years in the "Mayan long count" calendar, according to some theories. This calendar began at a date that is equivalent to 3114 BC in the Gregorian system; it counts days and has an integer value expressed in a 5-digit mixed-base system using bases 13, 20, 20, 18 and 20 in each of the five places. The rightmost two places, counting in cycles of 18×20, correspond to a 360-day year, and the other three places count "years" (periods of 360 solar days). The total number of days is 1872000. Expressed more accurately, the "5126 years" is 13×202×18×20/365.242189670 = 5125.3662719. See 2012.
The use of 13 as the base of the highest place in this calendar's counting system is uncertain, and it could just as likely be 20, which would lead to a "long count" of 18×204/365.242189670 or about 7885 years.
See also 18980.
Number of feet in a mile. 5280 is close to the Roman mile defined as 1000 paces, or 5000 feet, but was adapted to accomodate other units of length including the furlong (660 feet) and chain (66 feet). These numbers (66, 660 and 5280) are all multiples of 11 because of their relation to the old unit of length called a rod, also called pole, which in turn appears to go back to an old farmer's tool called an "ox goad", which happened to be about 11/2 yards long.
See also 1852.
6585.3213142 ≅ 223 × 29.530588853
Number of mean solar days in the saros.
6939.60160373 = 19 × 365.242189670
Number of mean solar days in the metonic cycle of 19 tropical years or 235 synodic months. Its integer approximation (6940) can be used as the basis of a lunisolar calendar that repeats every 19 years, and with some effort you can also make the months fairly regular. For example, every year can have 12 months each of which has a specific name, 6 with 29 days and 6 with 30 days; 7 out of 19 years have a 13th month of 30 days, and 4 of those 7 years also add an extra day to one of the normal 29-day months. Other similar systems are possible but all solutions have equal complexity; actual lunisolar calendars (like the Hebrew calendar) are more complex but achieve greater accuracy.
Although it is complex, a lunisolar calendar has very strong practical motivations. The importance of the solar day is obvious; the tropical year is important to anyone living in a climate with seasons. And the synodic month tells us when it is possible to see at night by the light of the moon, and what time of day the high and low tides will take place. These things are important even in modern urban society. Under a lunisolar calendar you can agree to meet outdoors at 8PM every month on the 15th of the month and know that there will be moonlight (weather permitting), or you can agree to go fishing every 10th of the month at 5AM and know there will be a low tide (assuming that's how the tide lines up at your location).
The integer approximation of 6939.60160373. There would be this many days in 19 years, if the 19-year metonic cycle were exact.
7920 is the ratio between 11!=39916800 and 7!=5040, and the ratio 7920/5040 = 11/7 = 1.571428571428... is kind of an approximation to π/2.
One of the more memorable small powers. While I was 10 and 11 years old I memorized integer exponents of integers just for fun. I still know all of these by heart:
| 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 | all the way up to | 131072 |
| 3 | 9 | 27 | 81 | 243 | 729 | 2187 | |||||
| 4 | see 2 | ||||||||||
| 5 | 25 | 125 | 625 | 3125 | 15625 | ||||||
| 6 | 36 | 216 | 1296 | 7776 | 46656 | ||||||
| 7 | 49 | 343 | 2401 | ||||||||
| 8 | see 2 | ||||||||||
| 9 | see 3 | ||||||||||
| 10 | trivial | ||||||||||
| 11 | 121 | 1331 | 14641 | ||||||||
| 12 | 144 | 1728 |
7776 is also a Kaprekar number for 5th powers: 77765=28430288029929701376, and 2843+0288+0299+2970+1376=7776.
(And better still, 7+7+7+6 = 27.)
7980 = 15×19×28, the product of three numbers that have an important relation to calendars in the Roman, Byzantine and Christian worlds. 28 years is the "solar cycle", 4×7, the number of years it takes before any date falls on the same day of the week again (that's 4 years per leap-year cycle, and 7 days per week). 19 is the length of the metonic cycle, see 19 for more. 15 years is the "indiction cycle", a period related to certain phenomena such as taxation. Since all three numbers are relatively prime to each other, the least common multiple is 7980. The "Julian day number", used in astronomy, is based on a system proposed in 1583 just after the adoption of the Gregorian calendar. Julian day 1 is January 1st 4713 BC. The year 4713 BC happened to be the most recent time that all three cycles (the 15- 18- and 28-year cycles) were aligned. It is convenient primarily because it pre-dates all recorded history, even in China, Egypt, Greece and Mesopotamia.
8000 = 203 = 113+123+133+143, the smallest cube that can be expressed as the sum of 4 consecutive cubes. See also 216.
This number has a property related to vampire numbers: if you take the first two digits and multiply by the other two, the product has the same four digits (but in a different order): 81×27=2187.
See also 8712.
(Personal: For a while during my childhood the numbers 7, 27 and 127 were my favorite 1- 2- and 3-digit numbers. I have since forgotten why the properties of 127 appealed to me, but I suspect it was partly because it ends in "27". If I had continued the series, I probably would have picked 8127, because it is divisible by 7 and by 27, and consists of the cubes 8, 1 and 27 strung together; also the "81" is 3×27.)
The smallest number that is not a palindrome, and is divisible by the number you get when you reverse its digits: 8712 = 4×2178. The next number like this is 9801=992. The numbers with this property comprise Sloane's A031877 and their factors are A008919. (Thanks to Tavi Laiu for this one). See also 1089. Another related property: 8712 × 2178 = 664.
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