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All numbers larger than 128 can be expressed as the sum of two or more distinct squares.
There no larger powers of two for which all of the digits are also powers of two.
128 is a power of two, and therefore vaguely related to computers; also the number of a state highway in the region of Boston, USA where a lot of computer companies got their start.
Gratuitous connection to 27: Make the 7 an exponent: 27 → 27 = 128.
132=11×12 is a multiple of 11 with 3 digits that uses each digit from 1 to 3 exactly once. 231=11×21 also satisfies this condition. With four digits, there are four solutions: 1243=11×113, 1342=11×122, 2431=11×221 and 3421=11×311. There is more about these numbers and their relation to an abstract type of "molecule" on this page.
Gratuitous connection to 27: 132=102+25 and 27=10×2+2+5.
The moon's phases occur on the same day of the year and the same day of the week every 133 years. 133 is 7 × 19, which is the number of days in the week multiplied by the number of years in the metonic cycle.
Gratuitous connections to 27: 1×33 = 27.
135 = 11 + 32 + 53, a cute arithmetical coincidence. Another example is 89 = 81 + 92. After the single-digit numbers, such numbers are fairly rare: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676, 2427, 2646798, 12157692622039623539, ... (Sloane's A032799). See also 153.
Gratuitous connection to 27: 13.5+13.5=27.
137 is ... uh, let's see ... the prime number closest to the average of 127 and 143! But perhaps it is more notable for being almost the reciprocal of the Sommerfeld fine-structure constant.
This is a recent (July 2008) value for the fine-structure constant (or more precisely, its reciprocal), originally called the Sommerfeld fine-structure constant and often referred to by the greek letter alpha. It is a dimensionless constant in physics. The expression "(51)" at the end of the value "137.035999084(51)" represents the "error range" in the value. This puts the actual, unknown precise value "probably" within the range 137.035999033 to 137.035999135. This is from Gabrielse75 and Hanneke78, March 2008. A previous value by Gabrielse with collaborators was 137.035999070(98).
Generally (although not at present, early 2009) the best known value would be found at CODATA, the authoritative source on such things. As of early 2009 they still have a July 2006 figure, 1/137.035999679(93). From 2002 to 2006 their value was 1/137.03599911(46); and from 1998-2002 it was 1/137.03599883(50). In all cases the CODATA figure is the reciprocal 0.00729735..., which is more relevant to its use; expressing this constant as 137.03... goes back to Eddington (as discussed below).
History
In the 10-15 year period following Einstein's development of general relativity, much work was done to try to unify the theories of electromagnetism, quantum mechanics and relativity. The fine-structure constant showed up in many formulas modeling electromagnetic phenomena, and it always showed up as a unitless expression involving several other already-accepted physical constants. For example, it was shown to be the ratio between the speed of the electron in its orbit in a classical Bohr atom and the speed of light. It is called the "fine-structure constant" because applying general relativity to the Bohr atom model explains the "fine structure" of the lines in hydrogen's spectrum, and the precise value of the electron's speed determines the width of the bands in the spectrum and other more easily measurable phenomena.
Using the methods of Hughes and Kinoshita the value has been computed quite accurately; the process includes experimental measurement combined with numerical integration of a large number of functions describing many different virtual particle interactions (each with a distinct Feynman diagram). It can be approximated with first-order, second-order, third-order, etc. approximations, according to how many virtual event pairs take place. The 1st-order approximation involves just one Feynman diagram, with a single event pair (in which the electron emits and reabsorbs a photon). Its calculation can be done on one page in about 1/2 hour. For a 2nd order approximation there are 7 diagrams; for 3rd order there are 72. The current best approximation (which is 4th order) took "many supercomputers over more than 10 years"50 to evaluate 891 diagrams. Each "order" reduces the error in the calculation by a factor of about 137, i.e. the current best estimate is off by about 1 part in 1374. These numbers (1, 7, 72, 891, 12672, ...) are Sloane's sequence A005413. It and other related sequences (with lots of example Feynman diagrams) are discussed in the 1978 paper by Cvitanovic, et. al.76.
Some have suggested that the constant may vary over time as the universe evolves. This can be tested by close measurement of the relative isotope abundances in a natural nuclear fission reactor, such as the one in Oklo, Gabon, Africa. This was attempted by two Los Alamos scientists77. They initially (in 2004) concluded that the constant may have decreased by 45 parts per billion over the last 2 billion years. However they have since refined the model and it is now once again likely that the constant is actually constant.
The Fine Structure Constant holds a special place among cult numbers: unlike its more mundane cousins 17 and 666, the Fine Structure Constant seduces otherwise sedate engineers and scientists into seeking mystical truths and developing uncollaborated theories. The fact that it is unitless, like π, seems to make people think that it should have some precise mathematical value. And, rather than embrace the idea that it is independent of the other unitless constants, they feel they need to express alpha in terms of some kind of formula, usually involving integers and functions that are taught in high school. The common problem is, in time more accurate experimental measurement results prove them wrong.
This began with scientists who already had quite a reputation, including Wolfgang Pauli, Karl Jung and Arthur Stanley Eddington. For a while Eddington worked on proving that 1/alpha was exactly 136 (see the entry on Eddington number); when measurements became more accurate he was compelled (by his own desire, not by others) to redo this work to "prove" it was 137.
In more recent years this tradition has spread to the larger community of science theory hobbyists. Mr. Michael Wales in the UK proposed that it is precisely the cube root of 2573380, or 137.0359896... based on the best available estimates at the time (which were third-order). A more elaborate and mystical approach is taken by Leahy (see the entry for 82944). Later, at Dr. James Gilson's web site www.fine-structure-constant.org he proposed this value for alpha:
29 cos(π/137) tan(π/(137×29)) / π
which gives 1/alpha = 137.0359997866991075..., which was within the 2002 CODATA error range but is now known to be well off the mark. Gilson's theory relates alpha to the length of the perimeter of a 137×29-sided regular polygon inscribed in a circle of unit diameter.
Another reader wrote to suggest the formula:
alpha ≅ e/(12 π3 me) = 0.0072972677066...
which gives 1/alpha ≅ 137.03759272566(52); the special value me is the ratio of the magnetic moment of the electron to the Bohr magneton, another unitless value given by CODATA as -1.00115965218111(74). The main problem with this formula is that it differs far more than experimental error allows: the value of 1/alpha is off by 0.000007, but less than 0.0000001 of this is accountable to the known errors in me and alpha.
The "simplest" formula I have seen so far was given by Steve Krakowski:
(37 / 0.27) - (1 / 964) = 137.0359996926386967880743...
which approximates the 2006 CODATA figure, and can also be stated as 137 + 1/27 - 1/964. Krakowski also gives the sequence:
(10(((666-72)/666)+3)/666)2 / (1+(10-3/(666))) = 137.036000...
(10(((666-72)/666)+3)/666)2 / (1+(10-3/(666-1))) = 137.035999692...
(10(((666-72)/666)+3)/666)2 / (1+(10-3/(666-3))) = 137.035999071...
which gives several old approximations of 1/alpha over the years, expressed mainly in terms of that other cult classic number 666, and the number 72 which appears in the sequence A005413 mentioned above.
For several years beginning around 1999, Mark Thomas (see also 3.377×1038) has been coming up with theories and formulas related to physics, and recently (late 2009) proposed a formula for alpha as well. His theory uses 702 and the Monster group size, among other numbers. (See his website here).
All this might not seem quite so amazing when you realize that there are many nifty formulas for any number. For example, my RIES program finds this in less than 4 seconds:
1/alpha ≅ ππe/2 + √(e3-1) = 137.035999746803...
(using the command ries -l3 -x 137.035999679). It was within the 2006 error bounds but is now known to be wrong. Now you need to search a little deeper, yielding this answer in about a minute:
1/alpha ≅ sqrt(((95+1)/π - 5sqrt(π)))
= 137.0359990922878250376708...
Many other fine structure constant theories are listed here.
In 2009 another reader (Dan Graham)79 suggested the formula π×e×C×13 / B, where C = sqrt(5)-1 = 1.236067977..., B = (1+A)(1/(1-A)), and A = 137/105. Graham's formula gives 137.03599906605045... so is within the current best known experimental error bounds. He expresses 13 as (52+1)/2, which gives the formula more symmetry by its similarity to C.
In 2004 Hans de Vries45 described the following curious convergent iterative calculation. It is another candidate still in the running, and is particularly attractive because it involves Schwinger's constant alpha/2π. One begins with any approximation to alpha, then repeats the following calculations as many times as desired:
G = 1 + alpha(1 + alpha/2π(1 + alpha/(4π2)(1 +
alpha/(8π3)(1 + ...)))))
alpha' = G2/(eπ2/2)
which converges on 1/alpha = 137.0359990958297... For automatic calculation one may wish to rewrite the G expression as
G = 1 + alpha + alpha2/2π + alpha3/(2π)3
+ alpha4/(2π)6 + alpha5/(2π)10 + ...
where the exponents of 2π are the triangular numbers. Here it is in Hypercalc's BASIC interpreter:
C1 = old fine-structure
C1 = list
10 scale = 20; a = 0.007;
20 for x = 1 to 11;
25 t = 0; g = 0;
30 for i = 0 to 5;
40 g = g + a^i/(2*pi)^t;
50 t = t + i;
60 next i;
70 a = g^2/(e^(pi^2/2));
80 fsc = 1/a
90 next x
99 end
C1 = run
R1 = fsc: 137.11712418125116745
R2 = fsc: 137.03717654594675111
R3 = fsc: 137.03601619522281466
R4 = fsc: 137.03599934415588632
R5 = fsc: 137.03599909943602158
R6 = fsc: 137.03599909588207334
R7 = fsc: 137.03599909583046107
R8 = fsc: 137.03599909582971153
R9 = fsc: 137.03599909582970065
R10 = fsc: 137.03599909582970049
R11 = fsc: 137.03599909582970048
To get more digits, double each of the scale and loop limits (from 20, 11 and 5 to 40, 22 and 10 respectively).
After learning that 32+42=52 and 33+43+53=63, one might wonder if the pattern continues in a similar way to 4th powers. But it doesn't: 34+44+54+64=74-143. See 8000.
143 is 11×13, a product of twin primes. Numbers with this property are: 15, 35, 143, 323, 899, 1763, 3599, 5183, ... (Sloane's integer sequence A037074). This is a more specific case of a number that is a product of consecutive primes.
The last 143 digits of 143143 (or, in other words, 143143 mod 10143) is a prime number. (That's not really such a big deal, unless you consider how much effort it took someone to figure it out!)
The Minot lighthouse in Massachusetts blinks in a 1-4-3 pattern: once, then four times after a pause, then three times after another pause, then a longer pause before repeating. Seamen (or their wives on shore) would see the light and imagine it was a message from their beloved: "i love you" = 1 letter, 4 letters, 3 letters.
The legend of Aladdin and the other stories of the Arabian Nights might not be as well-known if they had been called "143 Arabian Weeks".
Gratuitous connection to 27: 101.43143 ≅ 27.004. See also MCS01. ralimon ラリモン
144 is 122, and also happens to be the 12th Fibonacci number by the standard definition. See also 61917364224.
One of the largest numbers, that is not a power of 10, that has a specific word (gross) assigned to it. Moser is another.
Gratuitous connection to 27: 144 = (3+3+3+3)×(3+3+3+3); rearrange the parentheses to get 3+3+3+(3×3)+3+3+3 = 27.
150 = 2×3×52. It has no prime factors larger than 5, and this makes it a 5-smooth number. 5-smooth numbers include the 3-smooth numbers plus: 5, 10, 15, 20, 25, 30, 40, 45, 50, 60, 75, 80, 90, 100, 120, 125, 135, 150, 160, 180, 200, 225, 240, 250, 270, 300, 320, 360, 375, 400, 405, 450, 480, 500, 540, 600, 625, 640, 675, 720, 750, 800, 810, 900, 960, ... That list is Sloane's A080193 and is generated by multiplying all the 3-smooth numbers by 5, 25, and other higher powers of 5. The complete list of 5-smooth numbers is Sloane's A051037.
153 = 5!+4!+3!+2!+1!.
153 = 13+53+33. It is the smallest number that is the sum of powers of its own digits, where the power is the same as the number of digits, aside from the trivial 1-digit cases like 8=81. The next few numbers with this property are: 370, 371, 407, 1634, 4150, 4151, 8208, 9474, 54748, ... (Sloane's A023052). See also 4679307774 and 115132219018763992565095597973971522401.
153 = 17+16+15+...+3+2+1, the 17th triangular number. It is also 100+28+25, and appears in a New Testament story as a number of fish. A lot has been made of this in connection with the Enneagram, a system of personality type classification.
Gratuitous connections to 27: 153/(1×53) = 27; or 153/(1×5×3) = 225 and 22+5 = 27.
158 is the sum of row 7 of the "fibonomial triangle". This is a triangle of numbers similar to Pascal's triangle, using Fibonacci factorials in place of normal factorials. The numbers in the triangle are called fibonomials:
These numbers are defined similarly to the binomial coefficients in Pascal's triangle but using the Fibonacci numbers F1=1, F2=1, F3=2, F4=3, etc. (more here). For example, the 4th element in the 8th row (260) is F7F6F5/F3F2F1 = 13×8×5/2×1×1. The general form is Fn...Fn-k+1/Fk...F1. It is not immediately obvious that this formula always gives an integer. It does — because of the property of fibonacci numbers that if A is divisible by B, Fv∀ is divisible by FB, combined with the fact that the ordinary binomial coefficients use the formula {n}...(n-k+1)/k...1, which itself is always an integer for somewhat simpler reasons.
On Pascal's triangle, the second number on each row is the sequence of integers; here they are the Fibonacci number. On Pascal's triangle the following number is an oblong number; here the 3rd items (1, 2, 6, 15, 40, 104, 273, ...) are golden rectangle numbers. The next numbers after that (1, 3, 15, 60, 260, 1092, ...) are Fn×Fn+1×Fn+2/2.68,69
The same transformation can be done again to create the meta-fibonomial triangle.
163 appears in the "Ramanujan constant" epi×√163, which is very nearly an integer. It is the largest Heegner number, a set of 9 integers that share this same property. See 262537412640768743.999999... for a description of this and some related amazing numbers.
163 = 1 + 2 × 3 4 can be produced on most "algebraic" calculators by keying 1 + 2 × 3 xy 4 =. If you have an RPN calculator you would key in either 1 2 3 4 xy × + or 4 3 2 1 + × xy (depending on how the xy key works). See also 101.0979×1019.
Gratuitous connection to 27: 16310 = 6127 and 6110 = 2727 — or another way of saying the same thing: 2×27+7 = 61, and 6×27+1 = 163
173.297929 ≅ 29.530588853 × 223 / 38
The number of mean solar days between each time the nodes of the Moon's orbit align with the sun. This is the length of time between each eclipse season (the time of year when lunar and/or solar eclipses occur). The oldest known calendars of the Sumerians of Mesopotamia were based on the synodic month and the eclipse season (which was called by the same name that later became used for the solar year). Near the equator, the tropical year does not matter too much, but the moon was very important inasmuch as it provides light at night and eclipses were considered a very important event. The eclipse season is a little less than half a year, because the tilt of the moon's orbit keeps shifting; after 18 years it comes around to the same time of year again.
The magic constant of a 3×3 magic square that contains all prime numbers:
When the primes in this square are listed in order (5, 17, 29, 47, 59, 71, 89, 101, 113) a simple pattern can be seen: The successive differences are 12, 12, 18, 12, 12, 18, 12, 12. The values 12 and 18 don't matter so much as the symmetry of their arrangement.
See also 199.
Like 6, 12, 24, 48 and 96, 192 is a 3-smooth number of the form 3×2n, which causes it to be seen a little more often than other numbers of its size. It is also the first of several such numbers to occur in personal computer display dimensions (the Apple ][ hires graphics mode produced 192 rows of pixels); see also 768.
A member of the Lucas-Lehmer sequence defined by A0=4 and An+1=An2-2. This sequence starts 4, 14, 194, 37634, 1416317954, 2005956546822746114, ... (Sloane's A003010). It is used in a test to determine if a Mersenne number is prime; that test is described here. See also 47.
196 is a "4-dimensional pyramidal number" given by the sum: 6+5×22+4×32+3×42+2×52+62 = 6 + 20 + 36 + 48 + 50 + 36. This is similar to the formula for the tetrahedral numbers (for example, 6+5×2+4×3+3×4+2×5+6 = 56). It is because of this that squares stacked in 4-dimensions in a manner similar to the diagram here can be used to form a sequence of 4-dimensional pyramids corresponding to these numbers. There is also a general formula An-1 = n2(n2-1)/12. The sequence starts: 1, 6, 20, 50, 105, 196, 336, 540, ... (Sloane's A002415).
Take any number and reverse its digits, and add the two numbers together. For example, starting with 129, 129 plus 921 is 1050. Continue until you get a palindrome: a number which is equal to its own reversal: 1050+0501 = 1551, so 129 takes two steps. Some numbers take quite a while to arrive at a palindrome, and 196 is the first that goes so far that it is unknown whether it ever arrives at a palindrome. (In other bases it is possible to prove that certain numbers go on forever. In base 2 the number 2210 = 101102 produces an infinite sequence.)
199 is prime, and if you add 210 over and over again, you get 8 more primes. These 9 primes together can be arranged into a 3×3 magic square, and is the smallest possible magic square made of primes in arithmetic progression:
See also 177.
199 is also a Woodall number: 2×102-1.
The reciprocal 1/199 exhibits a pattern similar to that of 1/998. Here are the digits, broken into groups to show the structure; the repeating pattern is 99 digits long:
1/199 = 0.005 025 125 628140703517587939698492462311557788944
7236180904522613065326633165829145728 64 32 16 08 04 02 01 005 025 125
6281407...
Since 1/199 = 5/995, we have the digits of 1/995 (which contain powers of 5, for the reason explained under 998) multiplied by 5, which just means that the 0th power of 5 does not appear. The powers of 2 also appear, in the opposite order at the end, for reasons originating in the fact that 2×5=10 and we are working with base 10 digits.
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