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110

A number I have occasionally heard referred to as "eleventy". Most well-known is the "eleventy-first" birthday party for Bilbo in Tolkien's Lord of the Rings (By the way, there were several organized events honoring Tolkien's own eleventy-first birthday on Jan 3rd, 2003). I have also heard "eleventy billion" used as a jocular term for zillion.

111

111 is the next repunit after 11. A repunit is simply a number that consists of the digit 1 repeated. The term repdigit is used to refer to these plus numbers like 666 and 999999 that consist of some other digit repeated. These numbers were first discussed in Albert Beiler's 1966 book, Recreations in the Theory of Numbers (appropriately, in chapter 11). The shorthand R_n is used to refer to the repunit with n 1's. Repunits are the subject of a few conjectures and much recreational investigation. For example, which ones are prime? It is easy to show that R_n is prime only if n is prime — just divide by another repunit R_f where f is a factor of n. For example, R₄=1111 can be divided by R₂=11: 1111=101×11. The opposite need not be true — just because n is prime, does not mean that R_n is prime. For example, 111=3×37, and R₅=11111 = 41×271. It has been shown that R₂, R₁₉, R₂₃, R₃₁₇ and R₁₀₃₁ are the only prime repunits smaller than 10¹⁰⁰⁰⁰. That's about as high as we can go at present. In addition, R₄₉₀₈₁, R₈₆₄₅₃ and R₁₀₉₂₉₇ are considered probable primes. See also 12345654321 and 2.25573...×10¹⁵⁵⁹⁹.

111 is the atomic number of the element below gold on the periodic table. Given the nickname eka-gold (after Mendeleev's names for elements he predicted and were lated discovered), this element was first synthesized in 1994. Its most stable isotope has a half-life of 3.6 seconds. In 2004 it was officially given the name Roentgenium, after Roentgen, the physicist who discovered X-rays and received the first Nobel prize in physics. It is now possible to continue Hofstatder's amusing recursive story (pp. 103-126 of Gödel, Escher, Bach) one level deeper:

Meta-Meta-Genie: I'll have to send it through Channels, of course. One quarter of a moment, please... (And, twice as quickly as the Meta-Genie did, this Meta-Meta-Genie removes from the folds of his robe an object which looks just like the gold Meta-Meta-Lamp, except that it is made of Roentgenium, and it has "MMML" etched on it in even smaller letters, so as to cover the same area.)

Voice-of-Achilles: (another octave higher than before) And what is that?

Meta-Meta-Genie: This is my Meta-Meta-Meta-Lamp... (He rubs the Meta-Meta-Meta-Lamp, and a huge puff of smoke appears. In the billows of smoke, they can all make out a ghostly form towering above them.)

Meta-Meta-Meta-Genie: I am the Meta-Meta-Meta-Genie. You summoned me, O Meta-Meta-Genie? What is your wish?

[and so on...]

See also 82.

112

The hundredweight is a unit of weight equal to 112 pounds. I am told (by Chris Cotter) that sometime around the 1300's, this name was given to a unit of weight equal to 1/20 of a ton, which is called a "quintal" or a "zentner" in other languages. Since a ton (also called long ton for clarity) was 2240 pounds, the quintal is 112. There is also an "old" hundredweight of 108 pounds. The name has "hundred" in it because it is kind of close to a hundred; abbreviations for hundredweight include the letter C from the Roman numeral for 100.

120

120 is a value of (3ⁿ-3)/2, (Sloane's A029858; my MCS1979): 3, 12, 39, 120, 363, 1092, 3279, 9840, 29523, 88572, 265719, 797160, 2391483, 7174452, ... These numbers are sometimes involved in weighing problems (see 121) and are the denominators in the fractions discussed here.

120 is the smallest K-fold-perfect number for K=3. This is a number whose divisors add up to 3 times the number itself: 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360 (By contrast, for an ordinary perfect number, the divisors add up to twice the number: 1+2+3+6=12.) See also 30240.

120 = 2×3×4×5 = 4×5×6, the smallest number that can be expressed as a product of consecutive integers in two distinct ways. See 720 for more on this, also 210, 5040, 175560, 17297280 and 19958400.

121

121 is 11², and also a sum of powers of 3: 81+27+9+3+1 = 121. All the numbers from 1 to 121 can be expressed by adding or subtracting these 5 powers of 3 — for example, 58 = 81-27+3+1. 121 is a member of the series (Sloane's A003462; my MCS979) that you get by adding the first n powers of 3: 1, 4, 13, 40, 121, 364, 1093, 3280, 9841, 29524, 88573, 265720, 797161, 2391484, 7174453, 21523360, 64570081, ... In general each is (3ⁿ-1)/2 for some value of n.

There are a couple old math problems involving weighing that involve numbers of the form (3ⁿ-1)/2 (or in a few cases, (3ⁿ-3)/2 or (3ⁿ+1)/2). One such problem states, "how many weights do you need to be able to weigh things up to 40 ounces on a two-pan balance?" (The answer is 4 weights, of 1, 3 9 and 27 ounces). Another states, "how many weighings do you need to determine which one of 13 coins is of a different weight, if you know for certain that only one is different?" (The answer is three weighings; you start by comparing two groups of 4.⁴¹)

121 is an illustration of the binomial theorem. The digits of 121 are the same as the 3^rd row of Pascal's triangle.

125

This is 5³. The powers of 5 are: 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, ... . If you learn these you also learn the powers of 1/2, because 1/2^N = 5^N/10^N: 1/2 is 0.5, 1/4=0.25, 1/8=0.125, 1/16=0.0625, and so on. Starting with 25 they all end in "25", and the alternation of 125/625 continues indefinitely. The last 4 digits repeat in a cycle of 4: 0625, 3125, 5625, 8125. The last 5 digits repeat in a cycle of 8, the last 6 repeat in a cycle of 16, and so on. Patterns like this can be used to determine the trailing digits of certain huge numbers (see for example 27²⁷²⁷).

127

127 is 2⁷ - 1, an example of a number of the form 2^P - 1 where P is prime. All such numbers are called Mersenne numbers, but most are not prime. For example, the next Mersenne number after 127 is 2¹¹ - 1, which is 2047, but 2047 = 23 × 89, not a prime.

If a Mersenne number is prime (as is the case for 127) it is called a Mersenne prime. Here is a list of all known Mersenne primes.

127 also has the special property that 2¹²⁷-1 is also prime, see that entry for more.

Nth Term Sequences

Given a sequence S, you can create another sequence T by starting with T₀ as the first term of S, then T₁ = the T₀^th term of S, T₂ = the T₁^th term of S, and so on. The Ackermann function consists of sequences of this type.

127 is a member of the sequence you get when you do this using the prime numbers for S: The 1^st prime is 2, the 2^nd prime is 3, the 3^rd prime is 5, the 5^th prime is 11, and the 11^th prime is 31 — Continuing this process, you get: 127, 709, 5381, 52711, 648391, 9737333, 174440041, 3657500101, ... (Sloane's A007097) Each number T_n is approximately equal to ln(T_n-1) times T_n-1. This sequence grows slower than any exponential sequence A_n = Kⁿ for K>1, but faster than any fixed-power sequence B_n = n^K for K>1.

See also 8127.

128 = 2⁷

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.

133

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, the number of days in the week multiplied by the number of years in the metonic cycle.

135

135 = 1¹ + 3² + 5³, a cute arithmetical coincidence. 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.

137

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.

137.035999070(98)...

This is a fairly recent 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 "(98)" at the end of the value "137.035999070(98)" represents the "error range" in the value. This puts the actual, unknown precise value "probably" within the range 137.035998972 to 137.035999168. This is from Gabrielse, et. al. 2007⁴⁹. Generally (although not at present, early 2008) the best known value is found at CODATA, the authoritative source on such things; their figure was the best known since July 2006. Prior to 2006 (and since 2002) their value was 137.03599911(46).

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 1^st-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 2^nd order approximation there are 7 diagrams; for 3^rd order there are 72. The current best approximation (which is 4^th 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. The numbers (1, 7, 72, 891, 12672, ...) are Sloane's sequence A005413.

Some have suggested that the constant may vary over time as the universe evolves. This was tested by measuring relative isotope abundances at a natural nuclear fission reactor site in Oklo, Gabon, Africa.

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 sane engineers and scientists into seeking mystical truths and developing farfetched 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 algebraically independent of all other unitless constants, they feel they need to express alpha in terms of other constants and functions, usually involving integers and functions that are taught in high school. Their only problem is, more accurate experimental measurement results always prove them wrong. For a while Eddington tried to prove that it was exactly 136; when measurements became more accurate he was compelled to redo this work to "prove" it was 137. Recently 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). Most recently, at Dr. James Gilson's web site www.fine-structure-constant.org he proposes 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 m_e) = 0.0072972677066...

which gives 1/alpha ≅ 137.03759272566(52); the special value m_e 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 m_e and alpha.

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 ≅ π^πe/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. Many other fine structure constant theories are listed here.

In 2004 Hans de Vries⁴⁵ described the following curious convergent iterative calculation. It is the only 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/(e^π2/2)

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
If you want more digits, you should increase the scale and loop limits (20, 11 and 5) to proportionately larger values.

143

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: 10^1.43143 ≅ 27.004. See also MCS01. ralimon ラリモン

144

144 is 12², and also happens to be the 12^th 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

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

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 17^th 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

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: