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3121 = 5⁵-4

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

3456 is twice 1728, and has a notable pattern of ascending digits. It also has the following cute relationship to 27: 2⁷ × 27 = 3456.

3600

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.

4181

The first Fibonacci number with a prime subscript that is not itself prime: F₁₉=4181=37×113.

4879

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.

4900

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.

4979.003621... = e^(e(π-1))

This is e(_④)π, 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).

5040

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 reasons for 5040 being the ideal number of citizens in a state.

5041

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.

5280 = 2⁵×3×5×11

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 multiples of 11 because of 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).

6940

The integer approximation of 6939.60160373. There would be this many days in 19 years, if the 19-year metonic cycle were exact.

7776 = 6⁵

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:

7776 is also a Kaprekar number for 5th powers: 7776^5=28430288029929701376, and 2843+0288+0299+2970+1376=7776.

8000

8000 = 20³ = 11³+12³+13³+14³, the smallest cube that can be expressed as the sum of 4 consecutive cubes. See also 216.

8127

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.)

8712 = 2³ 3² 11²

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.

10000 = 10⁴

From the 3^rd century BCE to as late as the 12^th century, in Greek, Coptic and Armenian texts, letters of the alphabet are sometimes used to represent numbers⁴⁸:

Α=1 Β=2 Γ=3 Δ=4 Ε=5 ςσ=6 Ζ=7 Η=8 Θ=9 Ι=10 Κ=20 Λ=30 Μ=40 Ν=50 Ξ=60 Ο=70 Π=80 Ϙ=90 Ρ=100 Σ=200 Τ=300 Υ=400 Φ=500 Χ=600 Ψ=700 Ω=800 Π=900  ͵Α=1000  ͵Β=2000  ͵Γ=3000  ͵Δ=4000  ͵Ε=5000  ͵Δ=6000  ͵Ζ=7000  ͵Η=8000  ͵Θ=9000

The necessary letters are combined to make a quantity, so for example the number 8127 would be  ηρκζ. For numbers above this they used a capital letter mu (Μ) to represent 10000, whose name in Greek is myriad. To distinguish it from Μ=40, a small alpha was written above the Μ. The precise way in which larger numbers were handled differed through the centuries in ancient Greece. In the most ambitious system, a number (like  ηρκζ) was written in small letters directly above the M to represent 10000 raised to that power; see 10⁴⁰⁰⁰⁰. However, in much more common usage the small number placed above the M merely multiplied the value; see 10⁸.

Myriads and powers of a myriad are also involved in Chinese and other east-Asian systems for naming large numbers; see 10⁴⁰⁹⁶.

See also 100000 and 10000000.

10080

10080 = 60 × 24 × 7, the number of minutes in a week. 10080 is also a factor record-setter with 72 factors. Thus, there are 72 different ways to divide a week into equal parts that are whole multiples of minutes. 10080 is also 2×7!, see 40320 and 604800.

In France just after the revolution they switched to the now-standard system of weights and measures based on powers of 10 (SI, systeme internationale, or the "metric system"). They also set up (for a while) a calendar involving a 10-day week, and dividing the day into powers of 10. It didn't last long, and France switched back to the Gregorian calendar a few years later.

But there have been other times since then that people have considered the idea of measuring time in powers of 10. Clearly this "metric time" idea has some appeal, because there are dozens of web sites about it (do a search for "metric time"). Perhaps best-known is Swatch's "Internet time", based on the division of the day into 1000 parts called "beats" (with 86.4 seconds per "beat"). For subdivisions of the day it is easy, the hours and minutes we have now are just arbitrary divisions and there is no conflict with the other important environmental cycles.

When considering a metric replacement for the synodic month and tropical year, clearly there is no good practical solution. These are all very important physical cycles and have strange ratios that cannot be changed (at least not yet :-) and cannot easily be adapted to work with powers of 10.

Personally, I think the existing standard time system is just as cool as any 10-based system would be, because of the utilitarian properties of the factor record-setters and cute things like the minutes in February and the 86400000 property. However, if you insist on setting up a "metric time" system, you could do worse than to take advantage of the fact that a minute and a week are almost at an exact ratio of 10000. Such a system would leave the definition of "week" unchanged and define a special "minute" which is exactly 7/10000 of a day (which comes out to exactly 60.48 "standard" seconds). Then all weekly events would always start at the same minute-mark, which would be a 4-digit number from 0000 to 9999. You'd still have to deal with the fact that the number of weeks per year is odd (the fact that it's close to 100/2 makes this a little more bearable), the number of minutes per day is odd, and there is no suitable "month".

See also 3628800 and 86400000.

10201 = 101²

The 2^nd row of Pascal's Triangle shows clearly in the 2^nd power of 101. The pattern continues in higher powers, and goes further than for the powers of 11. See 14641 for more.

11811

The name of the worker who is befriended by Freder in the 1927 movie Metropolis. Also a palindrome, and strobogrammatic.

14641 = 11⁴

The smaller powers of 11 (121, 1331 and 14641) give rows of Pascal's Triangle. Pascal's Triangle is a rather useful table of numbers (called binomial coefficients) that is formed by continually adding numbers in a cascade starting from a single 1, as shown here:

Each is the sum of the (one or) two numbers above it.

These numbers have many applications because they count "combinations", such as "how many combinations of 3 colors can you make if you have 5 colors to choose from?" (the answer is the 3rd number in row 5 of the triangle).

A general formula that gives the value of item N in row M is:

_NC_M = M!/((M-N)!×N!)

For example, ₃C₅ = 5!/((5-3)!×3!) = 120/(2×6) = 10. However this formula gets rather impractical to use when M is large, even when the value being computed is pretty reasonable. For example, consider ₃C₁₄₃=143!/(140!×3!). 143! has 248 digits, and 140!×3! has 242 digits — so you probably couldn't even get the answer on a calculator, as the calculator would overflow. However, most of the 143! in the numerator cancels out with the 140! in the denominator, leaving ₃C₁₄₃ = 143×142×141/3! = 477191, which you could even calculate by hand if you needed to.

19683 = (3³)³ = 27×27×27

This is 3_④3 or 3^^3 by the lower (left-associative) version of the hyper4 operator. See also 196883.

22026.465794806... = e¹⁰

e¹⁰ is the highest value on the LL scale of most log-log slide rules that have such a scale. (However, a few go as high as 10¹⁰.)

25772.1300

As of 2003, the best known value of the Platonic Year, or precession cycle, in mean solar years (the error is ±0.0015 years — less than a day!). This is the time it takes for the orientation of Earth's poles to go one full circuit in its circular path. The movement is caused by the tidal influence of the Moon and Sun (and to far lesser extent, other planets like Venus and Jupiter) on the Earth, which has a gyroscopic precession effect because of the Earth's bulging a bit around the equator. The value is approximated as 25600, 25800, 25920 or 26000, depending on the purpose and source. The raw data value from which this is derived is the "general precession in longitude" in computer models, 5028.7955 ± 0.0003 arc-minutes per Julian Century⁸. A Julian Century is 36525 mean solar days, so the precession period is computed by (360 × 3600 × 100 / 5028.792) × (365.25 / 365.242189670).

25800

An often-seen approximation to the precession cycle.

25920 = 6×12×360

According to those ancient astronomers who knew of the "precession of the equinoxes" (now known to be movement of the Earth's axis of rotation, see 25772.1300), the ascending and descending nodes of the Sun move 1 degree through the zodiac every 72=6×12 years. This produces a cycle of 25920 years for the entire period. This isn't too far off from the modern value. An "age" of 2160 years, ¹/₁₂ of the full period, is also given much significance. See also 2592.

27000 = 30×30×30 = 3³×10³

Approximately the number of days in an expected lifetime for most readers of this page (and 75 "Babylonian" years, see 405000).

31337

A prime number, and one of many misspellings of the word "elite" used by those who wished to hide their conversations from automatic detection on bulletin-board systems in the 1990's. This "language" possibly evolved partly from inverted calculator-display text (see 71077345).

30240

30240 = 2⁵×3³×5×7, and is a 4-perfect number: Its divisors add up to exactly 4 times the number itself. Curiously, it also has the same digits, and most of the same factors, as 40320. See also 120, 154345556085770649600.

40320

This is 8! (8 factorial), 8×7×6×5×4×3×2×1, and is the number of minutes in 4 weeks. It would also be the number of minutes in a month, if months were exactly 4 weeks long. I'm no revolutionary — I'm not suggesting we should change the calendar — but I do strongly believe February should be designated "International Factorial Appreciation Month", to be observed in all years except leap-years. Alas, all my pleas have fallen on deaf ears (-: See also 10080, 604800 and 86400000.

In 1963 a set of bell ringers in Loughborough, Leicestershire, England became the first in history to ring eight tower bells in all possible permutations — each bell was rung 40320 times, for a total of 8×8! = 322560 blows, with each different combination of the 8 notes occurring exactly once. The feat took almost 18 hours⁴².

41616 = 204²

This is a square and also a triangular number: 41616 = 204² = 288×289/2. Such numbers are fairly rare: 1, 36, 1225, 41616, 1413721, 48024900, 1631432881, ... (Sloane's A001110). See 204 for more.

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