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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.
Since 120=5×24, 432000 seconds is exactly 5 days (see also 86400).
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. See also 10000000000 and 101010.
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. See also 7.525×1010.
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.
1969920 = 24×1080×76 = 28×34×5×19
The number of divisions of the day in the Talmudic (traditional Jewish law) measurement system used for astronomy, including the Hebrew calendar (see 29.5305941358):
| 1 רגע (moment) = 0.0439 seconds | |||
| 76 רגע | = 1 חלק | 1 חלק (part) = 3 1/3 sec | |
| 1080 חלק | = 1 שעה (hour) | ||
| 24 שעה | = 1 day |
|
| 1 רגעים (moment) = 0.26 seconds | |||
| 24 רגעים | = 1 עטים | 1 עטים (time) = 6.25 seconds | |
| 24 עטים | = 1 עונוט | 1 עונוט (period) = 2 minutes 30 seconds | |
| 24 עונוט | = 1 שעה (hour) | ||
| 24 שעה | = 1 day |
|
See 1001, 2001 and 6469693230.
This is 1283 and is also equal to 1253+503+253+153+33. That's not particularly special, but if you multiply everything by 4 you get the sum mentioned here.
The number of days in the "long count" cycle of the Mayan calendar (also used by the Aztec and Toltec cultures), a bit less than 5126 years. This cycle is 13 bak'tun, where a bak'tun is a period of 20 k'atun, a k'atun is 20 tun, a tun is 18 unial (pronounced wee nal), and a unial is 20 k'in or days. The unial is the same length as the Mayan "week", a period of 20 days each with a distinct name; 18 such periods plus the 5-day uayeb period of prayer and mourning made up the haab, or vague civil year. When this is combined with the 260-day tzol k'in, a larger cycle of 18980 days (52 haab) results, this is the least common multiple of 365 and 260, and thus is the number of days that elapses before the tzol k'in and civil year repeat with the same alignment. Mayans were also aware of the need for leap years. See also 260 and 2012.
2646798 = 21 + 62 + 43 + 64 + 75 + 96 + 87, a sum of its digits raised to consecutive powers51. See 135.
Last denominator in the greedy Egyptian fraction expansion of 4/17 = 1/5+1/29+1/1233+1/3039345. This is an example of why the greedy algorithm doesn't work too well.
This factorization record-setter is really easy to remember because it has 360 factors and includes two 360's in its digits, due to the fact that it is 360 times 1001 times 10. And its prime factorization is reasonably easy to remember if you learn it as 32×42×52 × 7×11×13; that should also help if you're trying to find its factors without using trial-and-error.
10! = 10×9×8×7×6×5×4×3×2×1 = 3628800 can also be expressed as 6!×7×6! because 6! = 720 = 10×9×8. Source: Gary Rosys See also 5040. 10! seconds is exactly 42 days; see also 10080, 604800 and 86400000.
There are no other factorials that can be broken up in this way. Call the middle number x (so, in the 10! case, x=7). Then the product of interest is (x-1)!×x×(x-1)!. The latter (x-1)! is also expressed of the form (x+1)×(x+2)×(x+3)×..., so we have:
1×2×3×...×(x-2)×(x-1) = (x+1)×(x+2)×(x+3)×...
The numbers (x+1), (x+2), etc. on the right side of the equation have to be composite, because if any of them is prime then they would not be a prime factor of the product on the left. The formula works for 7 because the numbers right after 7 (8, 9, 10) are not prime.
Also, all of the prime factors on the left side have to occur on the right hand side. Since the left side includes all the primes up to (x-1), all of those primes have to occur on the right. In the case of 7, the needed primes are 2, 3 and 5, which is not very many. Let p be the largest prime on the left. For example, if x is 13, p is 11. A multiple of p has to occur on the right, and the first multiple of p is 2p. This requires a fairly big gap in the distribution of primes. When x is bigger than 7, there will always be some other prime on the right side, that is greater than x and less than 2p. Proving this part is a little tough.
3628800 is also the number of seconds in 6 weeks: 6 × 7 × 24 × 60 × 60 = 6 × 7 × (8×3) × (3×4×5) × (10×2×3) = 6 × 7 × 8 × 3×3 × 4 × 5 × 10 × 2 × 3 = 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10. See also 86400000.
The number of ways to arrange a 2×2×2 Rubik's Cube. As there are no center cubelets to determine the orientation, one corner is considered to have a fixed, defined orientation. The other 7 can be put into any of the 7!=5040 possible positions, and all but one can be rotated into any of 3 different rotations (the total rotation of all 8 pieces always adds up to 360o).
See also 4.3252×1019, 7.4012×1045, 2.8287×1074, 1.5715×10116, and 1.9501×10160.
One in a series of crossover points in the level-index representation for numbers proposed by Lozier and Turner.
According to early Hindu mythology, the mahayuga or "great age" is a period of time consisting of four consecutive ages, lasting 1728000, 1296000, 864000 and 432000 years for a total of 4320000. They placed themselves and all of humanity in the fourth of these ages, see 432000. The great age repeats many times; the longer periods in the Hindu cosmological calendar are described under 622080000000000.
This number is an exceptional counterexample to the abc conjecture. The abc conjecture states that, given two relatively prime numbers a and b, the sum of the distinct prime factors of a, b and of their sum c=a+b, called rad(abc), is "almost always" bigger than c. For example when a=7 and b=33=27, c=34=2×17, which makes rad(abc)=2×3×7×17=714, quite a bit bigger than c. 6436343 is special because it is so far in the other direction: a=310×109, b=2, c=235=6436343, and rad(abc)=2×3×23×109=15042, much less than c.
8114118 is a palindrome, and the 8114118th prime 143787341 is also a palindrome. This is the smallest such number, aside from the trivial cases (like 11, the 5^{th] prime). The prime is a member of A046941 and its index is in A046942. It was discovered by Carlos Rivera35, and is followed by 535252535.
The first counterexample to the classical conjecture that any number of the form 2P-1(2P-1), with P prime, is perfect. See 2047 and 496.
A telephone number, subject of the popular song released in 1982 by the pop band Tommy Tutone. See also 525600, 10000000000 and 101010.
This is both a prime and a palindrome, the next-larger palindrome prime is 9136319. This would not be very special if it were not also for the fact that, in the digits of π, the digits 9136319 appear starting at position 9128219.
The first of a set of 5 consecutive primes that are spaced an equal distance apart: 9843019, 9843049, 9843079, 9843109 and 9843139 are all prime, there are no primes in between, and the spacing between each one and the next is 30. 9843019 is the lowest number with this property; the next is 37772429. See also 47, 251, 121174811 and 19252884016114523644357039386451.
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