Notable Properties of Specific Numbers
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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 A46941 and its index is in A46942. 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.
A unit of (Asian) Indian number name system. It is called crore when needed (primarily in Indian dialect of written English). In Iranian usage a crore is 500000. See also 10000 and 100000.
This is π④e, where ④ is the higher-valued form of the hyper4 operator. This value was computed using my generalization to real arguments based on the error function erf(x)). See also πe, 3581.875516... and 4341201053.37.
This is 224 and is equal to 2503+1003+503+303+63. Since all of those cubes except 63 end in 000, 216 shows up all by itself at the end of the number. See also 2097152 and 134217728.
A product of two non-overlapping sets of consequtive integers: 17297280 = 8×9×10×11×12×13×14 = 23×32×2×5×11×22×3×13×2×7 = 2×3×7×26×5×13×6×11 = 63×64×65×66. This type of match is is more "unlikely" than that demonstrated by 19958400 because it requires more prime factors to work out right after rearranging. See also 720, 175560, and Sequence A064224.
Combined fuel economy of a Toyota Prius, in SI units (50 miles per gallon converted to meters (of distance traveled) per cubic meter (volume of fuel consumed)).
See 3.1418708596056 and 137.035.
19958400 = 3 × 4 × (5×6×7×8×9×10×11) = (5×6×7×8×9×10×11) × 12 = 12! / 24. This is the prouct of the integers 3 through 11, and also the product of integers 5 through 12. There are an infinite number of ways to construct a number with this sort of pattern, all of which have a similar form: two consecutive numbers at the beginning (in this example 3×4) get replaced by their product, an oblong number (in this example 12), at the end. The general form is:
n×(n+1)×(n+2)×...×(n2+n-2)×(n2+n-1) = (n+2)×...×(n2+n-2)×(n2+n-1)×(n2+n) = (n2+n)!/(n+1)!
The sequence grows about as quickly as the factorials of the squares: 120, 19958400, 20274183401472000, 368406749739154248105984000000, ...
See also 17297280 and Sequence A064224.
The length (in meters) of the IUGC standard meridian. This represents the length of a line from one pole of the Earth to the other (crossing the equator midway, i.e. at about the 10,002-kilometer point). It is an international standard agreement, and is a sort of average of meridians at different longitudes75. The original definition of meter was based on the meridian and would have had this number be exactly 20000000. The original determination of the meter's length, based a massive seven-year surveying project, established a meridian length that was too small. Later improvements in understanding about the Earth's shape and extensive established use of the meter for non-surveying purposes made it necessary for the unit to diverge from its original meridian-based definition. The total change in length is about 195 parts per million. The meter ended up being a bit "shorter", so the meridian is now more than 20,000 km. See also 1852.
The last in a sequence of similar-looking prime numbers: 31, 331, 3331, ... are prime51. The following number is not: 333333331=17×19607843. See also 73939133.
The largest triangular number of the form T(x2-1) that is also 6 times another triangular number; see 91.
Lower bound for the number of states a 5-state, 5-tuple Turing Machine can make, on an initially blank tape, before halting, found by Buntrock and Marxen in 1990. See 107 for more.
Type this on a calculator and read the display upside-down; it (sort of) says "SHELL OIL". In the 1970's there were a bunch of joke "word problems" that instructed the reader to enter some sort of formula (example: 30 × 773 × 613 - 1 = × 5 =) to produce an answer that is read as a word by holding the calculator upside-down. For this purpose the digits 0,1,2,3,4,5,7,8,9 were used to represent O, I, Z, E, H, S, L, B and G respectively, so the answer/punchline could be any word or phrase using only these letters. See also 31337.
This number is prime, and if you take one or more digits off the end, the resulting numbers 7393913, 739391, ... 73, 7 are all prime. This is the largest number with this property. See also 33333331, 357686312646216567629137 and 3608528850368400786036725.
The number of milliseconds in a day: 86400000 = 24×60×60×1000. See also 10080, 40320, 432000 and 3628800.
The fifth hyperfactorial: 86400000 = 55×44×33×22×11. See also 55.
It seems rather odd that such a large number is listed for two unrelated properties, but there are larger examples (see 18446744073709551615).
An astronomical unit in miles; the approximation "93 million miles" was commonly taught in the US. The number is precisely defined by agreement, see here for details. See also light year.
A myriad myriad, and the largest number mentioned in the Bible (Hebrew תנ"ך (Tanakh) or Christian Old Testament): Daniel 7:10, "... and ten thousand times ten thousand stood before him, ..." (King James version). It is probably not a coincidence that 108 was also the largest number for which the Greeks had a name; the book of Daniel reached its final form well after Alexander conquered the entire Levant region. See also 666.
The first of a set of 6 consecutive primes that are spaced an equal distance apart: 121174811, 121174841, 121174871, 121174901, 121174931 and 121174961 are all prime, there are no primes in between, and the spacing between each one and the next is 30. 121174811 is the lowest number with this property; it was first discovered in 1967 by L. J. Lander & T. R. Parkin. Along with 2, 3, 251 and 9843019, forms a sequence (Sloane's A6560) that is thought to be infinite, but it is very hard to discover the next one. No one has yet discovered the first set of 7 consecutive primes; such a set would have to have a spacing of 210 or a multiple of 210; see 19252884016114523644357039386451. See also 47, 251 and 9843019.
This number, 227 or 233, is equal to this rather memorable sum of cubes: 5003+2003+1003+603+123. Another way to express this fact is:
ln((5322)3 + (5223)3 + (5222)3 + (3×4×5)3 + (3+4+5)3) = ln(2) 33
Scary but true: I actually discovered and verified this property of 227 by doing the math in my head. I already knew most of the powers of 2 up to 224=16777216. And, like tens of other kids around the world, I learned the squares up to 202 and the cubes up to 123 in grade school. One day I decided to double 224 a few times to get 227, then noticed the 217728, which looks a lot like 216 and 1728 stuck together. It was then fairly easy to see the rest, since 134 is 125 plus 8 plus 1. See also 2097152.
See also 151115727451828646838272.
The astronomical unit in kilometers. See 149597870691.
This number has 1008 distinct factors, and is the smallest number with at least 1000 factors. Its prime factorization is 26×32×52×7×11×13×17. See also 12, 840, 1260, 10080, 45360, 720720, 3603600, 278914005382139703576000, 2054221614063184107682218077003539824552559296000 and 457936×10917.
270270271 is prime, and is known to be a factor of 1010100+27. This seemingly amazing fact is actually quite easy to prove, using power-tower modulo reduction. Alpern 94 lists many such factors.
The smallest 9-digit number that, when written in three rows of 3 (as in one block of a Sudoku puzzle) forms a 3×3 magic square. There are 7 others: 294753618, 438951276, 492357816, 618753294, 672159834, 816357492, and 834159672.
The speed of light in meters per second. In 1983 by international agreement, the meter was redefined in terms of the speed of light, and as a result the constant for the speed of light is now exactly 299792458 meters per second. The second, in turn, is defined as precisely 9192631770 times the feequency of photons in a Caesium maser-based atomic clock. See also 2.54, 8.987552×1016, 1.6160×10-35 and 5.390×10-44.
The speed of light was first calculated from astronomical measurements in 1710 by Ole Romer, but had to be expressed as a ratio to the speed of Earth in its orbit (or equivalently, in terms of certain unknown Solar System distances and known light travel times) because the size of the astronomical unit had not yet been determined to sufficient accuracy; this would not come until the late 1700's (see 149597870691 for more).
A meter is also just about equal to the length of a pendulum with a period of precisely two seconds (a seconds pendulum, the length is close to 994 millimeters). In fact, this definition was proposed as the standard unit of length over 100 years before the original Metric system became official, and for most of the 18th century it was one of two competing proposals. The other proposal (based on the size of the Earth) was chosen because the period of a pendulum depends on where it is measured. (See 20003931.4585 for more about the meridian measurement and its errors).
It is a strange coincidence that the gravitational acceleration at Earth's surface (9.8 meters per second2) times the length of Earth's year (about 31557600 seconds) is about 310000000 meters per second, just a little bit bigger than the speed of light. There is no significance to this coincidence, it's just kind of cool. See also 3.14187.
See also 186282.397.
This is the smallest number that can be expressed as a×ba in three distinct ways: 344373768 = 8×98 = 3×4863 = 2×131222. See also 648.
This is the largest number you can express with just two digits and possibly one symbol (99, 9 ^ 9 or 9③9). See also 4.281248×10369693099 and 101.0979×1019.
456790123 has the "370-property": it is equal to the average of all possible permutations of its digits. Since there are 9 digits, there are 9! = 362880 permutatons. That would take a really long time to add up to take an average, but we can save a lot of work by noting that each digit occurs in each position an equal number of times. For example, the digit "4" will appear in each position in exactly 1/9 of the permutations. This effectively means that we can compute the average much more quickly just by using one representative permutation with each digit in each possible position. In this case, that can be done by computing:
(456790123 + 567901234 + 679012345 + 790123456 + 901234567 + 012345679 + 123456790 + 234567901 + 345679012) / 9
where the 9 terms are the original number rotated into all possible positions (like the multiples of 142857). If you take this sum (on a 10-digit calculator) you'll find that the average is equal to the original number, 456790123. These numbers of this type (first pointed out to me by reader Claudio Meller) are discussed more fully on their own page.
535252535 is a palindrome, and the 535252535th prime 11853735811 is also a palindrome. This is similar to 8114118 and was discovered by Giovanni Resta. The prime is a member of A46941 and its index is in A46942 35
The smallest number expressible as the sum of two 4th powers in two different ways: 635318657 = 592+1582 = 1332+1342. See also 50, 65 and 1729.
A billion in the "short scale" system used in the United States, and comparatively recently adopted by the UK and other English-speaking countries. Most other countries use the "long scale" in which a "billion" is 1012.
This difference in usage (109 versus 1012) came into being at a time when it didn't matter to most people. But thanks to many factors (population growth, inflation, prosperity, technology, and education) numbers in the billions are now very common in the news and in everyday speech. The honor associated with the name millionaire in the early 1900's now belongs to the billionaire. We often hear of costs and deficits in the billions; many of our computers have billions of bytes of storage capacity and perform billions of operations per second.
109 is an estimate of the processing power (in floating-point operations per second) embodied in a human retina. The retinas perform image processing to detect such things as edge movement and boundary direction. The figure is based on a resolution of roughly 106 pixels, a speed of 10 changes per second, and 100 FLOPs per pixel. See also 1018.
1000000001 = 11×90909091 = 1001×999001
Most of the numbers of the form 10n+1 can be factored in simple and pretty ways; this one happens to have two such factorizations.66 Here are most of the simpler patterns:
form examples
103n+1 1001=11×91 1000001=101×9901
1000000001=1001×999001
1000000000001=10001×99990001
105n+1 100001=11×9091
10000000001=101×99009901
1000000000000001=1001×999000999001
107n+1 10000001=11×909091
100000000000001=101×990099009901
102n+1+1 1001=11×91 100001=11×9091
10000001=11×909091
1000000001=11×90909091
104n+2+1 1000001=101×9901 10000000001=101×99009901
100000000000001=101×990099009901
106n+3+1 1000000001=1001×999001
1000000000000001=1001×999000999001
108n+4+1 1000000000001=10001×99990001
100000000000000000001=10001×9999000099990001
As you can see, there are two different sets of patterns. As long as n is a multiple of an odd number, 10n+1 fits at least one of the patterns. The numbers excluded by this are of the form 102i+1: 11, 101, 10001, 100000001, 10000000000000001, etc. (Sloane's A80176, the "base 10 Fermat numbers"). There is no easy factorization pattern for them.31
See also 1001.
A square in which each digit appears exactly once. (Contributed by Cyril Soler)
The Planck energy in Joules (kg-m/s2).
See 3.1418708596056 and 137.035.
3432948736 is the smallest number N such that N = 2N mod 10K, where K=10. In other words, 2 to the power of 3432948736 ends in the digits 3432948736. This is a member of a sequence (Sloane's A121319) that is thought to be endless. It has the nice property that each member of the sequence adds a digit to the previous one. For example, 28736 ends in 8736, 248736 ends in 48736, 2948736 ends in 948736, and so on.
4292853750 = 111111111101111110111111111101102
The Human population of the Earth according to the Arecibo message, which was transmitted in 1974. A more modern estimate is 6771000000. This is possibly the most dangerous number anyone has ever sent in any communication, because as Cassiday notes77, "Aliens who correctly interpret this will know how large an army to send".
See also 4294441822.
4294441822 = 111111111111011111111011010111102
Number of base-pairs in the Human genome, as given77 by the Arecibo message. A more modern estimate is 5941000000.
See also 4292853750.
First composite Fermat number. See here for more on these numbers; see also 17, 257, 641, (2222+1).
The number of years in the Hindu manvantara or "day of Brahma". See 1260 and 622080000000000.
This is e④π, where ④ is the higher-valued form of the hyper4 operator. This value was computed using my generalization to real arguments based on the error function erf(x)). See also eπ, 4979.003621... and 11058015.34616.
This number is equal to the sum of the 10th powers of each of its digits, and is unique in being the only 10-digit number to meet this requirement. Such numbers are called Armstrong numbers, Plus Perfect numbers, or narcissistic numbers. See also 153 and 115132219018763992565095597973971522401.
The number of base-pairs in a haploid human genome counting 46 chromosomes (23 from each parent) and assuming that there is one X and one Y chromosome (i.e. a male individual).
See also 3.01607×103576838408.
This is 29 primorial, 2×3×5×7×11×13×17×19×23×29 and has a really easy-to-remember digit pattern: 646 969 323 0. The pattern results from the properties of 1001=7×11×13 and 2001=3×667=3×23×29, which multiplied together give 2003001, and 323=17×19.
World population as of 2009 July 16th, as estimated by the U.S. Census Bureau, from the Wikipedia page. Another somewhat higher estimate is given by this site.
See also 1×1014.
Alternate answer to the "first prime number in alphabetical order" question (see 8018018851).
This is the first prime number in alphabetical order in the English language: "eight billion eighteen million eighteen thousand eight hundred and fifty-one". It was found by Donald Knuth. All other numbers that occur earlier in alphabetical order (like 8 and 8018018881) are composite.31
(Neil Copeland has suggested that 8000000081 comes earlier, based on the spelling "eight billion and eighty-one". Presumably Knuth leaves out the and.32)
The sixth perfect number. The even perfect numbers (it is not known if there are any odd perfect numbers) can all be expressed in the forms:
2P-1 (2P - 1)
2N (2N+1 - 1)
where P is a prime and N = P+1. In this example, P is 17. Also, for the number to be perfect, 2P-1 must be prime, and is called a Mersenne prime. See here for a complete list of known perfect numbers.
See also 4.4823309×1014471464.
Years in the Hindu "Day of Brahma" (see 622080000000000).
Frequency (in Hz) of microwave radiation used as the basis of the Caesium-133 atomic clock. This number is part of the official definition of the second (the basic unit of time).
The length of the second is originally derived from the rotation of the Earth and time-division decisions by the Babylonians, among other things (see 86400). Also, the rotation rate of the Earth keeps changing — it has changed by 19 parts per billion in the past 100 years49, enough to mean that this number could have been anywhere from about 9192631680 to about 9192631860 (and the number defining the meter and the speed of light, see 299792458, could have been anywhere from 299792455 to 299792461).
Ten billion. This number appears in a Schoolhouse Rock song; see 101010. See also 525600, 8675309, 100000000000 and 101010.
The upper limit of certain slide rule LL scales; see 22026.465794806.
The largest number that can be formed from the digits 1, 2 and 3 using the ordinary functions addition, multiplication and/or exponents. It slightly edges out 231=2147483648 because log(3)/log(2) is greater than 31/21. The next number in this sequence is 101.0979×1019.
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Quick index: if you're looking for a specific number, start with whichever of these is closest: 0.065988... 1 1.618033... 3.141592... 4 12 16 21 24 29 39 46 52 64 68 89 107 137.03599... 158 231 256 365 616 714 1024 1729 4181 10080 45360 262144 1969920 73939133 4294967297 5×1011 1018 5.4×1027 1040 5.21...×1078 1.29...×10865 1040000 109152051 101036 101010100 — -- footnotes Also, check out my large numbers and integer sequences pages.
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