| Notable Properties of Specific Numbers |
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Often related to 86400, 432000, etc. by people studying ancient and/or mystical things. Also, it has been noted that the great pyramid in Egypt has a perimeter of 3023.16 feet at its base, and estimated height (before erosion) of about 481.40 feet. If you multiply these by 43200 you get (almost exactly) the circumference of the Earth and its polar radius. See also 432, 432000, 4320000
45360 has 100 factors (including 1 and itself); no smaller number has as many. Its prime factorization is 24×34×5×7. To get one of the 100 factors of 45360, pick one number from each column in this table:
| A | B | C | D | |
| 1 | 1 | 1 | 1 | |
| 2 | 3 | 5 | 7 | |
| 4 | 9 | |||
| 8 | 27 | |||
| 16 | 81 |
|
and multiply them together to get a factor. Since you have 5 choices in each of the first two columns and 2 choices in the other two columns, the total number of choices you can make is 5×5×2×2=100. Notice that these numbers (5,5,2,2) are one more than the exponents in the prime factorization of 45360 (24345171). You can count the number of factors of any number just by taking the exponents in its prime factorization, adding one to each, and multiplying together. For 45360, the exponents are (4,4,1,1). Any other number with a prime factorization with two 4's and two 1's for exponents will have the same number of factors, including 2×34×54×13 = 1316250 and 2×5×74×114 = 351530410 and so on. Based on this, you can see that:
For more about factors record-setters, see 12, 60, 360, 840, 1260, 720720, 3603600, 245044800, 278914005382139703576000, 2054221614063184107682218077003539824552559296000 and 457936×10917.
The highest value of k for which 2k-1 + k is known to be prime, or is considered a "probable prime". The other lower values of k are 1, 3, 7, 237 and 1885. There ought to be about log(N) such k values for k < N, but since 2k grows so quickly, it is very difficult to find more. It also happens to be difficult to prove whether there is or is not an infinite number of such k's.
The largest finite value that can be represented in the 16-bit floating-point format "s10e5", also called "half" or "fp16", used in professional computer graphics. This format uses all the IEEE-754 rules with a 5-bit exponent and 10-bit mantissa. (More details here.)
The number of astronomical units in a light year. In my old Macintosh space flight program Orion, I used 216 as an approximation to this. See also 1.609344.
This is 216 = 224 = 2222, which is 2 ^^ 4 using the hyper4 operator. Also, since 2 ^^ 4 is 2 ^^ (2 ^^ 2), 65536 is 2 ^^^ 3 where ^^^ is the hyper5 operator.
Appears in Zork and several later Infocom adventure games:
The usage of the number probably originated from the MIT hacker culture74; see 69.
This number has a few interesting properties. First of all, it is (8×9×4)2 (or 82×92×42), the square of the product of its odd-numbered digits starting from the left. Only 1 and 784 share this property. Also, if you multiply these three (1×784×82944) the answer is ((1×2×3×4×5×6×7×8×9)/(1+2+3+4+5+6+7+8+9))2.
These, and properties of other numbers (notably including 37 and an older approximation of the fine structure constant) have been linked in many ways by Mr. D. G. Leahy as part of what he calls a "new universal consciousness".
The number of seconds in a day.
In Biblical time, the number of weeks (approximately) in the time before the Flood. The Bible lists ten people during this period, of whom Adam and Noah are the 1st and 10th, and gives the ages of each when the next was born. According to the oldest known version of that account (which is in the Hebrew Torah and most Christian bibles but differs from the Septuagint and Samaratan versions) the number of years before the flood was 1656 (Genesis 5:3 to 5:29 and 7:11: Adam 130; Seth 105; Enos/Enosh 90; Cainon/Kenan 70; Mahalalel 65; Jared 162; Enoch 65; Methuselah 187; Lamech 182; Noah 600); this is almost exactly 86400 weeks (you have to fudge the math a bit: assume every 23 years has 5 leap years; then every 23 years would be exactly 1200 weeks; 1656=23×72 so there are 1200×72=86400 weeks.
Regarding the link to 432000: since the Babylonian calendar had 5-day weeks, 86400 weeks would be 432000 days in the Babylonian calendar. Under 432000 there is a description of the Babylonian flood story (Gilgamesh) and a period of 432000 years, which (as in the Biblical version) is expressed in terms of ten people who lived during the period prior to the flood.
The Babylonian story is older and it is generally acknowledged (by anthropologists, theologically neutral historians and such) that the Bible version is descended from it. It has been proposed that the sum was preserved and transformed, as the story traveled from the Babylonian to the Hebrew culture, in the following way: 432000 years became 432000 days, which then became 86400 5-day weeks, which was then reinterpreted as 86400 7-day weeks, which became 1656 years. This explanation works well because only the last step need be approximate, and the exact answer (86400 days = 1655.84 years) rounds up to 1656.
A unit of (Asian) Indian number name system. It is called lakh when needed (primarily in Indian dialect of written English), a name that derives from Sanskrit laksha. See also 10000 and 10000000.
This number has a property which exemplifies some of the many, obscure and somewhat arbitrary investigations into number theory that can be explored by anyone with the interest (and perhaps a personal computer). 140800 has the property that when expressed in each of 8 bases, the product of its digits equals the base times the sum of its prime factors. (The bases are all huge, and range from 198 to 2720.) There are many other similar (and some even more obscure) bits of number theory discussed on the Internet on Kevin Brown's math pages. See also 239.
The decimal expansions of the fractions 1/7, 2/7, 3/7, 4/7, 5/7, and 6/7 all consist of the digits 142857 repeated. Notice the factorization: add 7 and you get 33×7×11×13×37 = 999999, which is of course related to the facts that 33×37=999 and 7×11×13 = 1001.
A Kaprekar number: 1428572=20408122449, and 20408+122449=142857.
175560 is a rather remarkable example of a product of two non-overlapping sets of consequtive integers: 19×20×21×22 = 19×22×5×3×7×2×11 = 5×11×23×7×3×19 = 55×56×57. See 720 for other examples; see also 17297280 and Sequence A100933.
Richard Borcherds won the Fields Medal (the math equivalent of the Nobel prize) in 1998 for his proof of an amazing coincidence involving this number.
Some years after the Monster group was discovered, it was shown that the minimum number of dimensions of a crystal lattice whose symmetry rotations and reflections form the Monster group is 196883. It was then noticed that this is only one less than the number 196884 that occurs in the elliptic modular function responsible for the Ramanujan constant and connected to the proof of Fermat's Last Theorem.
There was much speculation about whether these two numbers 196883 and 196884 were actually related for a reason. It turns out the answer is yes — and for showing this Borcherds won the Fields Medal.
See also 19683.
A term in a polynomial that comes up in relation to the so-called Ramanujan constants. See also 196883.
This is 603+123, more artistically expressed as (3×4×5)3+(3+4+5)3. See also 134217728.
1260 years is approximately the lifetime of an eclipse in a saros series.
Numerology knows no limits. While researching 217728 I found the following false claim: "In Revelation Chapter 12 there are five numbers; 12, 24, 3, 1296, and 2, which when multiplied together total 217728." 217728 is actually 12×24×3×126×2, so at first it looks like the numerologist missed the 9 key on his calculator as he was entering 1296. (Not hard to believe, as he goes on to say that "217728 also equals 81 times 2268"; it is actually 81×2688.) But, if you actually read Rev 12, you find that the numbers mentioned there are (in order) 12, 7, 10, 7, 1/3, 1260, 2; there is also a "half" in Rev 12:14. You may or may not choose to count the fractions; our numerologist seems to have counted "a third" in Rev 12:4 as a 3. Apparently he decided to combine the 7+10+7 into 24 (perhaps because they occur in the same verse), and dropped the 0 on 1260 to make 126. That would at least make the product equal 217728, but the basic claim still isn't satisfied. If he had done it "right" his product would have to be 12×7×10×7×(1/3)×1260×2 = 4939200, or 14817600 if you don't count the 1/3 as a number, or 44452800 if you count it as a 3, and various other non-217728 values if you add a 1/2 or 31/2 from Rev 12:14 (see 1260 for more on Rev 12:14).
For more about numerology, see 666.
This is another way to define an exponent-based factorial function (the best-known version is the hyperfactorial). This version grows much faster because the height of the "tower of exponents" keeps growing: 1, 21=2, 321=9, 4321=262144, 54321 = 6.206069882×10183230, 654321 = 10(4.829261049×10183230), ...
The first of a set of 111 consecutive composite numbers: Every number from 370262 through 370372 is composite. This is the first time there are 100 or more composite numbers in a row. The preceding prime, 370261, and the following prime 370373, together set a record for largest gap between primes. For every number of digits, there are about 4 primes that set a record like this. The sequence is: 2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, ... (Sloane's A002386). See also 1693182318746372.
The ratio between broad daylight and a clear night under a full moon. It is a rather impressive feat of nature that the human eye can handle this range so well. The expression "2.51214" refers to the differences between the Moon's and Sun's apparent visual magnitudes (which is 14) and the ratio represented by one unit on the logrithmic magnitude scale.
The nimesha, 1/405000 of a day, is a division of time in one (of
several) Hindu system of measurement. This one is notable for
involving consecutive powers of 30. (this one
uses powers of 60).
| 1 nimesha = 213 msec | |||
| 15 nimesha | = 1 kashita | 1 kashita = 3.2 sec | |
| 30 kashita | = 1 kala | 1 kala = 96 sec | |
| 30 kala | = 1 muhurta | 1 muhurta = 48 min | |
| 30 muhurtas | = 1 ahoratra | 1 ahoratra = 1 day | |
| 30 ahoratra | = 1 month |
|
The nimesha breaks the pattern of powers of 30, but sometimes a unit of "half-nimesha" is used (see 2202). As I have shown, one can also extend the system in the other direction by adding the 30-day month, which is the more commonly-used month in Hindu calendars. Continuing further you get 900 days (roughly 2.5 years), 27000 days (73.92 years, which is curiously just about a lifetime), and 810000 days, 2217.7 years or a little longer than the time for precession to go from one zodiac sign to the next. See also 26244000000, 4665600000000.
432000 years is also 120 sars (a sars is 3600 years), the length of the kali-yuga, which in Hindu mythology is the length of the age in which the early writers placed themselves. There are several longer lengths of time in the same mythology; see 4320000.
A Babylonian version of the flood story (analagous to that in Genesis) tells of ten kings who ruled during the period prior to the flood; the total length of that period is 432000 years. (See 86400 for a link to the Biblical version.)
In pre-Christian Germanic mythology, there is sort of a reference to the number 432000: 800 men at each of 540 gates of Wodan's palace.
Similarly, in the Viking doomsday tale of the Day of Ragnorook, 800 divine warriors emerge from each of 540 doors of Valhalla.
See also 432, 43200, and 4320000000.
453600 = 360×1260
Number of days in a Biblical prophetic period (see 3.5 and 1260). It is also 10 times the highly-composite number 45360.
510510 is a primorial: 510510 = 2×3×5×7×11×13×17. It is also the product of four consecutive Fibonacci numbers: 510510=13×21×34×55. The digits repeat because 1001 is the product of the 3 consecutive prime numbers 7×11×13.
Number of minutes in a year, if you take a year to be exactly 365 days. This number is repeated several times in the song "Seasons of Love" in the stage musical Rent. At 11 syllables, it is the longest number name that I know of that has been set to music. It beats out 8675309 by three syllables.
604800 = 60×60×24×7, the number of seconds in a week. It is also 10!/6. See also 10080, 40320, 86400 and 86400000.
640320 = 2001×320 = 26×3×5×23×29
One of the numbers that comes up in relation to the Ramanujan constant, also notable for being easy to remember.
It takes this many Hebrew years for the Hebrew calendar to repeat itself. The factor of 19 is the metonic cycle, 7 is the number of days in the week, and the other factors (a total of 26×34 = 5184) come from the units of division used to approximate the length of the synodic month (see 29.5305941358)
The number of atoms in a poliovirus, as modeled by the simulation program VirusX.
The first of about 20 successive factorization record-setters that have a repeating digit pattern. 720720 has 240 factors; the next factors record-setters are 1081080, 1441440, 2162160, 2882880, 3603600, and so on. The digits repeat because the numbers are multiples of 1001, and multiples of 1001 occur as factors record-setters because 1001 is the product of 3 consecutive prime numbers.
This is the number of days from the date given in Ezekiel 31:1 until September 11, 2001. The sum 1290 + 1335 is significant, see 1260.
Fermat's Little Theorem states that if p is a prime number and not a factor of n, then np-1 - 1 is divisible by p. When n = 10 and p = 7 we get 107-1 - 1 = 106 - 1 = 999999 which is divisible by 7. This gives us the first "interesting" repeating decimal fraction, 1/7.
If p is 2 the formula is trivial; in all other cases p is odd and we can express np-1 - 1 as:
np-1 - 1 = (n(p-1)/2 + 1) (n(p-1)/2 - 1)
which gives at least two other factors; in the case of 999999 these are 999 and 1001. By Fermat's Little Theorem, at least one of these must be divisible by 7 (in this case it's 1001).
See also 999.
One million. This number is probably associated with the concept "really big number" by more people than any other number (see also thousand and billion).
One million is at about the limit of direct physical perception. You can just barely put 1,000,000 dots on a large piece of paper and stand at a distance such that you can perceive each individual dot as a distinct dot, and at the same time be within viewing distance of the other 999,999 dots. (I have actually done this, just for fun!). Because it is near the limit of physical perception, I use 1000000 as the boundary between the class 1 and class 2 numbers.
The English name million comes from French (and Old French) milion, probably from Old Italian milione, which is a colloquial way of saying "big thousand".54
A near miss to Fermat's Last Theorem, 1419869 = 135+165 = 175+12. (And, you can remove the exponents — 13+16 = 17+12)
Fermat's Last Theorem
Fermat was reading the Arithmetica by Diophantus, and he conjectured that there are no solutions to the equation
An + Bn = Cn
for positive integers A, B and C and positive exponent n>2. He then wrote, in the margin of his copy of Arithmetica, his famous (or infamous!) comment:
Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere: Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet. (Latin: [For] a cube [to be the sum of] two cubes, or a 4th power [to be the sum of] two 4th powers, or in general [for] any [number that is a power] greater than a square [to be] divided into [a sum of] two like [powers], is not possible: I have discovered a truly remarkable proof which this margin is too small to contain.)
Thus began almost 400 years of speculation and intensive effort by the mathematics community to discover what, if any, idea Fermat might have had when he made such a claim, and more importantly, whether the "theorem" (actually a conjecture) was in fact true.
Certain exponents (values of n) are easy to prove. For example, if the exponent n was a product of primes n=pq, and if there was a solution An+Bn=Cn, then it could be re-written (Ap)q+(Bp)q=(Cp)q, or Dq+Eq=Fq where D=Ap, E=Bp, and F=Cp. Thus, if the "theorem" is proven for any exponent q, all multiples of q also get proven.
And Fermat himself wrote a proof that if x2+y2=z2 and x,y,z are positive integers, xy/2 cannot be a square. From this it follows that A4+B4=C4 has no solution in positive integers A,B,C. That leaves only the odd prime exponents. In the late 1700's, Euler (almost) proved the case n=3, but his proof had flaws which had to be fixed posthumously. The case n=5 was proven next, but it took until 1825 and the work of several leading mathematicians.
Work got progressively more complex; by 1847 all odd primes up to 31 had been proven; many more were proven by hand and then computer techniques were used to prove all primes up to 4000000. The actual complete proof came in 1993 and 1994 and was based on the study of elliptic curves, something seemingly unrelated to Fermat's Last Theorem.
There are of course many "near misses" to the theorem. The smallest and best-known is 103+93=123+1=1729.
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