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 360^{o}).
See also 4.3252×10^{19}, 7.4012×10^{45}, 2.8287×10^{74}, 1.5715×10^{116}, and 1.9501×10^{160}.
One in a series of crossover points in the levelindex 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. See also 8640000000.
This is 9^{7} and is also the sum of 27^{4} and 162^{3}. See also 512.
This is the "original" Smith number, and was in fact the telephone number of someone named Smith. A Smith number is a number for which the sum of the digits is equal to the sum of the digits of its prime factors: 4937775 = 3×5×5×65837, and 4+9+3+7+7+7+5 = 3+5+5+6+5+8+3+7. @numberphile has a video on it: 4937775  Smith numbers. See also 22.
The length (in meters) of the major (transverse) axis of the ellipsoid (or oblate spheroid) used by the WGS 84 model to approximate the shape of the Earth. This is very close to the average equatorial radius of the Earth, if you measure based on where the gravitational field is equal to that at sea level. (The sea, being a fluid, tends to equalize its height profile such that gravity is the same at all points on its surface, and the WGS 84 model is calibrated to agree with sea level as closely as possible). See the Geoid article for an explanation of how the geoid (the "gravitational equipotential surface") differs from the actual surface of the Earth. Apart from following the sea height as just mentioned, it tends to be underground below any significantly elevated land. Local changes of density in the mantle and crust add lots of variation.
If the earth were a sphere and the meter agreed exactly with its original definition, this would be exactly 20 million divided by pi.
See also 298.257223563 and 20003931.4585.
6436343 = 3^{10}×109+2 = 23^{5}
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=3^{3}=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=3^{10}×109, b=2, c=23^{5}=6436343, and rad(abc)=2×3×23×109=15042, much less than c.
8114118 is a palindrome, and the 8114118^{th} prime 143787341 is also a palindrome. This is the smallest such number, after a few early trivial cases (like 11 which is the 5^{th} prime). The prime is a member of A46941 and its index is in A46942. It was discovered by Carlos Rivera^{35}, and is followed by 535252535.
The first counterexample to the classical conjecture that any number of the form 2^{P1}(2^{P}1), with P prime, is perfect. See 2047 and 496.
The digits of the most iconic phone number in the history of 1980's onehit wonders. "8675309/Jenny" was recorded by pop band Tommy Tutone in 1981 and was on the charts for some months thereafter. For the xkcd version and a cool bonus, see 867.5309....
See also 525600, 10000000000 and 10^{1010}.
This is both a prime and a palindrome, the nextlarger 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.
See also 7427466391.
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.
10^{7} appears in the definition of the vacuum permeability constant μ_{0}, also called the "permeability of free space", in the curious formula:
μ_{0} = 4π/10^{7} N/A^{2}
where N is newtons and A is Amperes. Those are both longestablished units in the SI system, so one might wonder where this 10^{7} comes from.
A current formal definition of the ampere is "the constant current which will produce an attractive force of 2×10^{7} newtons per metre of length between two straight, parallel conductors of infinite length and negligible circular cross section placed one metre apart in a vacuum". There is that factor of 10^{7} again, right in the definition of the ampere. Persuing this further goes right back to the definition of μ_{0}, a circular definition.
To reveal the origin of the 10^{7} we have to look at the history of the ampere unit and the discovery of the force between two electric wires carrying current, a phenomenon first demonstrated by AndreMarie Ampere in 1820. The (historical) original definition of the modern unit is 1/10 of the unit now called an abampere, which in turn was "the amount of current which generates a force of two dynes per centimetre of length between two wires one centimetre apart". A dyne is a g cm/s^{2}, and a newton is a kg m/s^{2}, so a dyne is 10^{5} newtons. In the units that were common in Ampère's time, μ_{0} was simply 4π:
μ_{0} = (4π/10^{7}) N / A^{2}
= (4π N) / (10^{7} A^{2})
= (4π 10^{5} dyne) / (10^{7} (abampere/10)^{2})
= (4π dyne) / (10^{2} (abampere/10)^{2})
= 4π dyne / abampere^{2}
So we see that the 10^{7} in the modern definition of μ_{0} is a relic of the old centimetre gram second system of units. Converting from dynes to newtons diminished the value by 10^{5}; measuring the force per meter of wire rather than per centimeter increased the value by 10^{2}; moving the wires from a distance of 1 cm to 1 meter canceled that 10^{2} out, and measuring the current in amperes rather than abamperes reduced the force by 10^{2} because Ampère's force is proportional to the product of the currents in the two wires and both measurements change by a factor of 10 (the number of amperes in an abampere).
10^{7}=10000000 is a unit of the (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 π^{④}e, where ^{④} is the highervalued form of the hyper4 operator. This value was computed using my generalization to real arguments based on the error function erf(x)). See also π^{e}, 3581.875516... and 4341201053.37.
This is 2^{24} and is equal to 250^{3}+100^{3}+50^{3}+30^{3}+6^{3}. Since all of those cubes except 6^{3} end in 000, 216 shows up all by itself at the end of the number. See also 246924, 2097152 and 134217728.
A product of two nonoverlapping sets of consecutive integers: 17297280 = 8×9×10×11×12×13×14 = 2^{3}×3^{2}×2×5×11×2^{2}×3×13×2×7 = 2×3×7×2^{6}×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 xkcd 687 and 3.1418708596056; see also 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 product 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)×...×(n^{2}+n2)×(n^{2}+n1) = (n+2)×...×(n^{2}+n2)×(n^{2}+n1)×(n^{2}+n) = (n^{2}+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 10,002kilometer point). It is an international standard agreement, and is a sort of average of meridians at different longitudes^{75}. 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 sevenyear 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 nonsurveying purposes made it necessary for the unit to diverge from its original meridianbased definition. The total change in length of the meter through this process was about 195 parts per million. The meter ended up being a bit "shorter", and the initial meridian measurement was too short (by a greater amount), so the average meridian is now known to be nearly 20,004 km. See also 1852.
The number of seconds in common (nonleap) year: 365×86400. Although Leap seconds are called "intercalary", they are effectively part of the year because the leap second occurs during the day, local time (for example, in a time zone 7 hours away from UTC, the clocks would go from "16:59:59" to "16:59:60" to "17:00:00") so a common year with a leap second would be 31536001 seconds long.
See also 432000, 604800, 31556952 and 31622400.
The number of SI seconds in a tropical year, according to xkcd 1061. The "Earth Standard Time" system, which is "simple, clearly defined, and unambiguous", defines a year by the following rules:
1 year = 12 months; 1 month = 30 days; 1 day = 1440 minutes (= 24
hours 4 minutes); 1 minute = 60 SI seconds. This gives 1 year =
31190400 SI seconds.
For 4 hours every full moon, run clocks backward. Full moons
happen every synodic month, which is 29.530588853×86400.001 =
2551442.9... SI seconds. After going backwards for 4 hours, the clocks
have to go 4 hours forward before continuing, so 8 hours = 8×3600 = 28800
seconds will be added. This increases the average length of a year by
a ratio of (2551442.9...+28800)/2551442.9... = 1.01128773..., making it
31542468.8... SI seconds.
The nonprimenumbered minutes of the first full nonreversed
hour after a solstice or equinox happen twice. The 17^{th} prime number
is 59, so there are 43 nonprime minutes in an hour. There are two solstices
and two equinoxes per year, so this rule adds 43×60×4 = 10320 seconds
to the year, for a year of 31552788.8... SI seconds.
31556925.6... = 365.242189670 × 86400.001
An approximation of the number of SI seconds in a mean tropical year, as experienced during the years 19952012. This is based on the average rate of rotation of the Earth during that period (see mean solar day) combined with the tropical year length for the year 2000 (which is in mean solar days). The figure for the length of the mean solar day has less precision because of the variations in Earth rotation rate on short timescales^{49},^{125}, due to weather and ocean currents, etc. whereas the year length figure represents an average over a period of several years.
The approximation to the number of seconds in a mean tropical year used in the 1956 and 1960 definitions of the SI second :
the fraction 1/31,556,925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time.
This number is related to Newcomb's solar motion coefficient as:
31556925.9747 × 129602768.13 = 36525×86400 × 360×60×60
In words, the number of SI seconds in the mean tropical year multiplied by the Sun's mean rate of motion in arcseconds per century is equal to the number of seconds in a Julian century times the number of arcseconds in a full circle.
See also 129602768.13.
This is the number of seconds per year according to the Gregorian calendar (averaged over a 400year period): 365.2425 times 86400. It is an exact integer but is just an average; the number of seconds in any particular year is always either 31536000 or 31622400.
Randall Munroe[211] found the approximation 75^{4}=31640625, which is a better approximation than the popular (among physicists) π×10^{7} = 31415926.535... .
Number of seconds in a Julian year (often used in astronomical ephimerides, for things like proper motion of stars, orbital elements of planets, etc.).
The number of seconds in a leap year: 366×86400. Although Leap seconds are called "intercalary", they are effectively part of the year because the leap second occurs during the day, local time (for example, in a time zone 7 hours away from UTC, the clocks would go from "16:59:59" to "16:59:60" to "17:00:00") so a leap year with a leap second would be 31622401 seconds long.
See also 432000, 604800, 31536000 and 31556952.
The last in a sequence of similarlooking prime numbers: 31, 331, 3331, ... are prime^{51}. The following number in the series is not: 333333331=17×19607843. See also 73939133.
The largest triangular number of the form T_{(x21)} that is also 6 times another triangular number; see 91.
Lower bound for the number of states a 5state, 5tuple Turing machine can make, on an initially blank tape, before halting, found by Buntrock and Marxen in 1990. See 107 for more.
This is the number of different ways that one can visit the state capitols of the 48 contiguous states in the United States, passing through each state only once. The same route in reverse does not count as a distinct route, and one end of the trip must be in Maine because it only borders one other state. The answer, and a description of algorithms used to calculate it, are in Knuth [161] section 7.1.4 (Binary Decision Diagrams), (p. 255 in the 2011 edition).
See also 25623183458304.
Type this on a calculator and read the display upsidedown; 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 upsidedown. 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, 381654729, 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 = 5^{5}×4^{4}×3^{3}×2^{2}×1^{1}. 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).
This is the smallest (positive) integer expressible as the sum of two (positive) cubes in three different ways: 167^{3}+436^{3} = 228^{3}+423^{3} = 255^{3}+414^{3} = 87539319. See also 1729.
An astronomical unit in miles, calculated using the IAU definition of the former. The approximation "93 million miles" was commonly taught in the US. This 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 10^{8} 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.
10^{8} is 億 in China (yì, dàng) and Japan (oku), where they construct numerals on the basis of 10, 100, 10000, 10^{8}, and higher powers of 10^{4}. This system closely resembles the Knuth yllion naming system for very large powers of 10. See my article on large numbers in Japanese
The number of DNA base pairs in the genome of the nematode worm C. elegans. See 959.
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.
129600000 = 100 × 360 × 60 × 60.
Number of arcseconds in a circle times 100. See 129602768.13.
Newcomb's coefficient giving the average rate of motion of the Sun across the sky (or equivalently, the rate of Earth's motion in its orbit, relative to the stars) in units of arcseconds per century. One might think this number should just be 129600000, but the Earth's axial precession and other effects prevent this.
See also 31556925.9747.
This number, 2^{27} or 2^{33}, is equal to this rather memorable sum of cubes: 500^{3}+200^{3}+100^{3}+60^{3}+12^{3}. Another way to express this fact is:
ln((5^{3}2^{2})^{3} + (5^{2}2^{3})^{3} + (5^{2}2^{2})^{3} + (3×4×5)^{3} + (3+4+5)^{3}) = ln(2) 3^{3}
Scary but true: I actually discovered and verified this property of 2^{27} by doing the math in my head. I already knew most of the powers of 2 up to 2^{24}=16777216. And, like tens of other kids around the world, I learned the squares up to 20^{2} and the cubes up to 12^{3} in grade school. One day I decided to double 2^{24} a few times to get 2^{27}, 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, based on the IAU definition of the former. See also 149597870691.
In early 2009, one David Horvitz (an artist who enjoys posting unusual ideas on his blog) suggested that people should take a photo of themselves standing in front of a fridge or freezer with the door open and their head in the freezer, then share it online (e.g. with Instagram or Flickr) tagged with the number 241543903. The idea caught on (becoming an internet meme) and an image search for this number will now return dozens of such photos.
See also 3112066128.
This number has 1008 distinct factors, and is the smallest number with at least 1000 factors. Its prime factorization is 2^{6}×3^{2}×5^{2}×7×11×13×17. See also 12, 840, 1260, 10080, 45360, 720720, 3603600, 278914005382139703576000, 2054221614063184107682218077003539824552559296000 and 457936×10^{917}.
270270271 is prime, and is known to be a factor of 10^{10100}+27. This seemingly amazing fact is actually quite easy to prove, using powertower modulo reduction. Alpern ^{94} lists many such factors.
The smallest 9digit 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.
299128000
An approximation to the speed of light hypothesized to be in Sayana's commentary on the Rigveda; see 2202 for details. See also 309467700.0.
and the definition of the meter (length unit)
299792458 is 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 frequency of photons in a Caesium maserbased atomic clock. See also 2.54, 8.987552×10^{16}, 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 18^{th} 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 second^{2}) 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.
The mean acceleration due to gravity on the Earth's surface, times the number of seconds in a mean tropical year. This happens to be only a few percent larger than the speed of light. This serves as a guideline to some basic limits on longduration manned space flight. Since astronauts would probably need to experience no more than about 1.1 or 1.2 times normal gravity during their trip, it would take a few years (even from the astronaut's own relativistic frame of reference) to make the trip even to the nearest stars.
This is the smallest number that can be expressed as a×b^{a} in three distinct ways: 344373768 = 8×9^{8} = 3×486^{3} = 2×13122^{2}. See also 648.
This 9digit number contains one each of the digits 1 through 9, and has the additional property that the first two digits (38) are a multiple of 2, the first 3 digits (381) are a multiple of 3, and so on up to the whole thing being a multiple of 9. You can see a bit of symmetry in the digits: the first three digits (381) plus the last 3 (729) add up to 10×111, and the middle 3 (654) plus itself in reverse (456) also adds up to 10×111. This type of number is called polydivisible, and there are lots of such numbers if you don't care about having one each of the digits 1 through 9. See also 30000600003 and 3608528850368400786036725.
This is the largest number you can express with just two digits and possibly one symbol (9^{9}, 9 ^ 9 or 9^{③}9). See also 4.28...×10^{369693099} and 10^{1.0979×1019}.
456790123 has the "370property": it is equal to the average of all possible permutations of its digits. Since there are 9 digits, there are 9! = 362880 permutations. 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 10digit 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 535252535^{th} 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}
This is a power of 2, and a 9digit number in which all 9 digits are different. There is no 10digit power of an integer in which each of the digits 0 through 9 appears once. See also 295147905179352825856.
The smallest number expressible as the sum of two 4^{th} powers in two different ways: 635318657 = 59^{4}+158^{4} = 133^{4}+134^{4}. See also 50, 65 and 1729.
The false Polya conjecture stated that positive integers with an odd number of prime factors always outnumber those with an even number of prime factors. In this case, the "number of prime factors" is sequence A001222, in which the same prime can be counted twice (so for example 8=2^{3}, 12=2^{2}×3 and 30=2×3×5 are all counted as having 3 prime factors). But the conjecture turns out to be false in a small region starting at 906150257 and extending up to 906488079.
See also 1.397162914×10^{316}.
See also 997002999.
A billion in the "short scale" system used in the United States, and comparatively recently adopted by the UK and other Englishspeaking countries. Most other countries use the "long scale" in which a "billion" is 10^{12}.
This difference in usage (10^{9} versus 10^{12}) 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.
10^{9} is an estimate of the processing power (in floatingpoint 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 10^{6} pixels, a speed of 10 changes per second, and 100 FLOPs per pixel. See also 10^{18}.
1000000001 = 11×90909091 = 1001×999001
Most of the numbers of the form 10^{n}+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:

As you can see, there are two different sets of patterns. As long as n is a multiple of an odd number, 10^{n}+1 fits at least one of the patterns. The numbers excluded by this are of the form 10^{2i}+1: 11, 101, 10001, 100000001, 10000000000000001, etc. (Sloane's A80176, the "base 10 Fermat numbers"). There is no easy factorization pattern for them. ([147] pp. 137138)
See also 1001.
A square in which each digit appears exactly once. (Contributed by Cyril Soler)
This is the decimal value of the hexadecimal integer constant 0x5f3759df that comprises the central mystery to the following bit of code, which is mildly famous among bitbummers and purports to compute the function f(x) = 1/√x:
float InvSqrt (float x) { float xhalf = 0.5f * x; int i = *(int*)&x; // evil floating point bit level hacking i = 0x5f3759df  (i>>1); // First approximation (WTF ?!?) x = (float)&i; x = x*(1.5f  xhalf*x*x); // Newton iteration // x = x*(1.5f  xhalf*x*x); // Iterate again if you need full accuracy return x; }This code actually works. It performs four floatingpoint multiplys, one floatingpoint add, an integer shift, an integer subtract, and two register moves (FP to Int and Int back to FP). It generates the correct answer for the function to within three decimal places for all valid (nonnegative) inputs except infinity and denormals.
The hex value 0x5f3759df is best understood as an IEEE floatingpoint number, in binary it is 0.10111110.01101110101100111011111. The exponent is 10111110_{2}, which is 190 in decimal, representing 2^{(190127)} which is 2^{63}. The mantissa (after adding the hidden or implied leading 1 bit) is 1.01101110101100111011111_{2}, which is 1.43243014812469482421875 in decimal. So the magic constant 0x5f3759df is 1.43243014812469482421875×2^{63}, which works out to the integer 13211836172961054720, or about 1.3211...×10^{19}. This is (to a firstorder approximation) close to the square root of 2^{127}, which is about 1.3043...×10^{19}. The reason that is significant is that exponents in 32bit IEEE representation are "excess127". This, combined with the fact that the "exponent.mantissa" floatingpoint representation crudely approximates a fixedpoint representation of the logarithm of the number (with an added offset), means that you can approximate multiplication and division just by adding and subtracting the integer form of floatingpoint numbers, and take a square root by dividing by two (which is just a rightshift). This only works when the sign is 0 (i.e. for positive floatingpoint values).
Here are some example values of numbers from 1.0 to 4.0 in IEEE singleprecision:
0.10000001.00000000000000000000000 = 4.0
0.10000000.10000000000000000000000 = 3.0
0.10000000.00000000000000000000000 = 2.0
0.01111111.10000000000000000000000 = 1.5
0.01111111.00000000000000000000000 = 1.0
Here I have shown the sign, exponent and mantissa separated by dots. Since the logarithm of 1 is zero, the value for 1.0 (0.01111111.00000000000000000000000) can be treated as the "offset". If you subtract this offset you get these values, which approximate the logarithm of each number:
0.00000010.00000000000000000000000 = 10.0_{2} = 2.0; log_{2}(4)=2
0.00000001.10000000000000000000000 = 1.1_{2} = 1.5; log_{2}(3)≈1.585
0.00000001.00000000000000000000000 = 1.0_{2} = 1.0; log_{2}(2)=1
0.00000000.10000000000000000000000 = 0.1_{2} = 0.5; log_{2}(1.5)≈0.585
0.00000000.00000000000000000000000 = 0.0_{2} = 0.0; log_{2}(1)=0
From this it is easy to see how a rightshift of the value for 4 yields the value for 2, which is exactly the square root of 4, and a right shift of the value for 2 gives the value for 1.5, which is a bit higher than the square root of 2. Over a full range of input values, the rightshift and addition of the magic constant gives a "piecewise linear" approximation of 1/√x.
The constant "0x5f3759df" is most commonly cited as being found in the Qrsqrt function of "game/code/qmath.c" in the source code of the videogame Quake III. It is attributed to John Carmack, but the same hack appears in several earlier sources going as far back as 1974 PDP11 UNIX.
David Eberly wrote a paper[170] describing how and why the approximation works.
Chris Lomont[173] followed up with investigation into its origins, getting as far as a claimed credit to Gary Tarolli of Nvidia. He thoroughly analyzes the piecewise linear approximation for odd and even exponents and proposes 0x5f375a86 as being slightly better, and a similar constant 0x5fe6ec85e7de30da for use with 64bit IEEE double precision.
David Eberly then wrote a longer explanation[200] analyzing the constant 0x5f3759df along with some other candidates (like 0x5f375a86 and 0x5f37642f). It describes efforts to discover why and how this value originally got chosen; with inconclusive results.
An earlier example of code calculating the square root in this way (approximation via a single shift, possibly with an add or subtract, no conditional testing, and Newton iteration) was described by Jim Blinn in 1997, where we find the following code: (see [159]).
inline long int AsInteger(float f) { return * (long int *)&f; } inline float AsFloat(long int i) { return *(float *)&i; } const long int OneAsInteger = AsInteger(1.0f); // 0x3F800000 float ASqrt(float x) /* Approximate Square Root */ { int i = (AsInteger(x)>>1) + (OneAsInteger>>1); return AsFloat(i); }with the comment:
This is actually pretty weird. We are shifting the floatingpoint
parameter — exponent and fraction — right one bit. The loworder bit
of the exponent shifts into the highorder bit of the fraction.
But it works.
 Jim Blinn ([159] page 83)
The same article discusses several similar functions including ones that include one iteration of Newton's method. Here are his inverse square root functions:
float AInverseSqrt(float f) { int i = (OneAsInteger + (OneAsInteger>>1))  (AsInteger(f)>>1); return AsFloat(i); } float BInverseSqrt(float x) { float y = AInverseSqrt(x); return y*(1.5.5*x*y*y); }If these are combined together into a single function with the inlines expanded, we get:
// OneAsInteger defined as above, equals 0x3F800000 const long int Magic = OneAsInteger + (OneAsInteger>>1); // 0x5F400000 float BInverseSqrt(float x) { int i = Magic  ((*(long int *)&f) >> 1); float y = *(float *)&i; return y*(1.5.5*x*y*y); }A much older example is found in the UNIX library sqrt function for the PDP11, dating back to June 1974 (see [139]):
/ sqrt replaces the f.p. number in fr0 by its square root. newton's method / ... movf fr0,(sp) asr (sp) add $20100,(sp) movf (sp)+,fr0 /initial guess / ...which is effectively performing an integer rightshift on the 16 high bits of the input value, then adding a constant similar to the constants in the above examples, and putting the result back into a floatingpoint register before proceeding with the Newton's method calculations. Only the upper part of the mantissa is being shifted, but that's good enough. A man page from Feb 1973 (Third Edition UNIX) suggests that the routine existed as early as then.
An alternate value (hexadecimal 0x5f375a86) of this magic constant.
Another alternate value (hexadecimal 0x5f37642f) of this magic constant.
The Planck energy in Joules (kgm/s^{2}).
See 3.1418708596056 and 137.035.
This number is associated with the UNIX epoch, which (on 32bit systems) will "roll over" on 2038 Jan 18^{th}. Numberphile has a video on it here: End of time (2147483647)
See also 49.710269... and 11644473600.
In late 2014 a Twitter friend and I undertook a challenge to find the smallest (integer, not starting with any 0's) number that does not appear in any Google search results (or, at the very least, try to estimate how many digits it would have).
Using Fermi Estimation (see Randall Munroe's whatif 84), I estimated that: there are 10^{10} people, each has 1 webpage, each with 1000 words; but only 1% of these are devoted to long lists of unique numbers (like invoice numbers, telephone numbers, etc.), and probably 90% of them are either small and duplicate each other somewhat, or are big and leave gaps. Answer: the smallest integer not indexed by Google is probably 10 digits long.
He and I spent a while trying numbers, and pretty quickly found that the 10digit numbers seem to be almost all taken. 11digit examples were easy to find. After just 10 minutes or so we had gotten down to the very low 11 digits (my best was 10826746091, his was 11170063270).
He kept looking for 10digit numbers, and noticed that there seem to be extensive lists of primes, but not of composites. He discovered that 6255626957 = 109×3803×15091 was unknown to Google, and soon after found that the Marshall Islands have telephone prefix 625 (thus offering a possible explanation). Shortly after this, he and another had found 3112066128 = 2^{4}×3×64834711.
Clearly this number will be indexed soon after appearing on this page, so I would call it a "likely upper bound" for whatever number is actually the smallest positive integer not in any Google result.
See also 241543903.
3432948736 is the smallest number N such that N = 2^{N} mod 10^{K}, 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, 2^{8736} ends in 8736, 2^{48736} ends in 48736, 2^{948736} ends in 948736, and so on.
4292853750 = 11111111110111111011111111110110_{2}
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 notes^{77}, "Aliens who correctly interpret this will know how large an army to send".
See also 4294441822.
4294441822 = 11111111111101111111101101011110_{2}
Number of basepairs in the Human genome, as given^{77} by the Arecibo message. A more modern estimate is 5941000000.
See also 4292853750.
4294967296 = 2^{32} = 2^{25} = 2_{④}6
The theoretical number of 32bit IP addresses; the actual number is a few percent lower because some values are reserved for special purposes. See also 281474976710656.
First composite Fermat number. See here for more on these numbers; see also 17, 257, 641, (2^{222}+1).
The number of years in the Hindu manvantara or "day of Brahma". See 1260 and 622080000000000.
This is e^{④}π, where ^{④} is the highervalued form of the hyper4 operator. This value was computed using my generalization to real arguments based on the error function erf(x)). See also e^{π}, 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 10digit number to meet this requirement. Such numbers are called Armstrong numbers, Plus Perfect numbers, or narcissistic numbers. See also 153 and 115132219018763992565095597973971522401.
(basepairs in the Human Genome)
The number of basepairs 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 3.01607×10^{3576838408}.
13 factorial, the number of way to rearrange 13 distinguishable objects. This number factors into some playingcard probabilities, such as 635013559600 and 2.235197...×10^{27}. See also 1716.
This is 29 primorial, 2×3×5×7×11×13×17×19×23×29 and has a really easytoremember 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 16^{th}, as estimated by the U.S. Census Bureau, from the Wikipedia page. Another somewhat higher estimate is given by this site.
See also 10^{14}.
The first 10digit prime number that appears as 10 consecutive digits of e:
e = 2.7182818284 5904523536 0287471352 6624977572 4709369995 9574966967 6277240766 3035354759 4571382178 5251664274 2746639193 2003050353 5475945713 8217852516 6427427466 ...
This is the answer to a puzzle that appeared on billboards in 2004. The billboards stated:
{first 10 digit prime in consecutive digits of e} . com
This little bit of nerd sniping led the solver to another, harder puzzle also involving digits of e. That puzzle, if solved, brought the user to a website soliciting resumes, potentially resulting in a call from someone at Google.
See also 9128219.
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 fiftyone". It was found by Donald Knuth. All other numbers that occur earlier in alphabetical order (like 8 and 8018018881) are composite. ([147] p. 15 footnote)
Neil Copeland has suggested^{32} that 8000000081 is the alphabetically first prime, based on the spelling "eight billion and eightyone". The use of and is common outside the U.S. (I have confirmed reports from the UK and New Zealand). Knuth, consistent with his statement in [143], does not use and.
See also 2.000...×10^{63} and 2.135987...×10^{96}.
8589869056 = 2^{16} (2^{17}1)
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:
2^{P1} (2^{P}  1)
2^{N} (2^{N+1}  1)
where P is a prime and N = P+1. In this example, P is 17. Also, for the number to be perfect, 2^{P}1 must be prime, and is called a Mersenne prime. See here for a complete list of known perfect numbers.
See also 4.4823309×10^{14471464}.
Years in the Hindu "Day of Brahma" (see 622080000000000).
(length of a second in Caesium133 units)
Frequency (in Hz) of microwave radiation used as the basis of the Caesium133 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 timedivision 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 years^{49}, 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 10^{1010}. See also 525600, 8675309, 100000000000 and 10^{1010}.
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 2^{31}=2147483648 because log(3)/log(2) is greater than 31/21. The next number in this sequence is 10^{1.0979×1019}.
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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 5×10^{11} 10^{18} 5.4×10^{27} 10^{40} 5.21...×10^{78} 1.29...×10^{865} 10^{40000} 10^{9152051} 10^{1036} 10^{1010100} — — footnotes Also, check out my large numbers and integer sequences pages.
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