© 1996-2008 Robert P. Munafo. Email the author
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| Notable Properties of Specific Numbers |
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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 |
|
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), amounting to about 5125 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.
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.
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.
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.
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 Rivera, and is followed by 535252535. 35
A telephone number, subject of the popular song released in 1982 by the pop band Tommy Tutone. See also 525600.
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). See also 10000 and 100000.
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 A100933.
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 A100933.
Number of meters in a meridian line from a pole to the equator. This is an international standard agreement, and is a sort of average of meridians at different longitudes. The original definition of meter was based on the meridian and would have had this number be exactly 20000000; 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 definition. 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.
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 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 and 357686312646216567629137.
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.
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, and 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 A006560) 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.
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 101010100+27. This seemingly amazing fact is actually quite easy to prove, using power-tower modulo reduction. There are
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. See also 2.54, 1.6160×10-35 and 5.390×10-44.
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.
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 101.0979465941272×1019.
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 A046941 and its index is in A046942 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. Most other countries use the "long scale" in which a "billion" is 1012. In 1974, the British government officially started using the smaller definition of "billion".
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 A080176, the "base 10 Fermat numbers"). There is no easy factorization pattern for them.31
See also 1001.
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© 1996-2008 Robert P. Munafo.
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