Mathematical coincidence

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A mathematical coincidence is said to occur when two expressions with no direct relationship show a near-equality which has no apparent theoretical explanation.

For example, there is a near-equality close to the round number 1000 between powers of 2 and powers of 10:

${\displaystyle 2^{10}=1024\approx 1000=10^{3}}$

Some mathematical coincidences are used in engineering when one expression is taken as an approximation of another.

Introduction

A mathematical coincidence often involves an integer, and the surprising (or "coincidental") feature is the fact that a real number arising in some context is considered by some standard as a "close" approximation to a small integer or to a multiple or power of ten, or more generally, to a rational number with a small denominator. Other kinds of mathematical coincidences, such as integers simultaneously satisfying multiple seemingly unrelated criteria or coincidences regarding units of measurement, may also be considered. In the class of those coincidences that are of a purely mathematical sort, some simply result from sometimes very deep mathematical facts, while others appear to come 'out of the blue'.

Given the countably infinite number of ways of forming mathematical expressions using a finite number of symbols, the number of symbols used and the precision of approximate equality might be the most obvious way to assess mathematical coincidences; but there is no standard, and the strong law of small numbers is the sort of thing one has to appeal to with no formal opposing mathematical guidance. Beyond this, some sense of mathematical aesthetics could be invoked to adjudicate the value of a mathematical coincidence, and there are in fact exceptional cases of true mathematical significance. All in all, though, they are generally to be considered for their curiosity value or, perhaps, to encourage new mathematical learners at an elementary level.

Some examples

Rational approximants

Sometimes simple rational approximations are exceptionally close to interesting irrational values. These are explainable in terms of large terms in the continued fraction representation of the irrational value, but further insight into why such improbably large terms occur is often not available.

Rational approximants (convergents of continued fractions) to ratios of logs of different numbers are often invoked as well, making coincidences between the powers of those numbers.

Many other coincidences are combinations of numbers that put them into the form that such rational approximants provide close relationships.

Concerning π

• The first convergent of π, [3; 7] = 22/7 = 3.1428..., was known to Archimedes, and is correct to about 0.04%. The third convergent of π, [3; 7, 15, 1] = 355/113 = 3.1415929..., found by Zu Chongzhi, is correct to six decimal places; this high accuracy comes about because π has an unusually large next term in its continued fraction representation: π = [3; 7, 15, 1, 292, ...].
• A coincidence involving π and the golden ratio φ is given by ${\displaystyle \pi \approx 4/{\sqrt {\varphi }}=3.1446\dots }$. This is related to Kepler triangles. Some believe one or the other of these coincidences is to be found in the Great Pyramid of Giza, but it is highly improbable that this was intentional.
• There is a sequence of six nines in pi that begins at the 762nd decimal place of the decimal representation of pi. For a randomly chosen normal number, the probability of any chosen number sequence of six digits (including 6 of a number, 658 020, or the like) occurring this early in the decimal representation is only 0.08%. Pi is conjectured, but not known, to be a normal number.
• ${\displaystyle {\frac {5}{6}}\pi \approx \varphi ^{2}}$; correct to within 0.002%.
• ${\displaystyle {\frac {1}{2}}\cdot {\frac {3}{4}}\cdot {\frac {5}{6}}\cdot {\frac {7}{8}}\cdot \dots {\frac {13}{14}}\cdot 15\approx \pi }$; correct to within 0.05%.

Concerning e

• The number 1828 repeats twice in a row after the 2nd digit in the decimal expansion of e and immediately after that the next digits are 459045, which show the angles of an isosceles right triangle in degrees. After that, the next 3 digits are the first 3 prime numbers, and after that is 360, the sum of all angles of a square/circle. e = 2.7 1828 1828 45 90 45 2 3 5 360....
• Related to the above, ${\displaystyle e\approx 2721/1001}$. When calculating e in base 60 or 120, it is useful to find 1001 (1+1/1!+1/2! .. +1/13!). These are exact expressions in these bases. Also, 718 + 281 = 999, (i.e. 718 281 718 ... is 719/1001), which makes this one of the convergences to e.
• There is a sequence of "99 999 999" within the first 500,000 digits of e.

Concerning base 2

• The coincidence ${\displaystyle 2^{10}=1024\approx 1000=10^{3}}$, correct to 2.4%, relates to the rational approximation ${\displaystyle \textstyle {\frac {\log 10}{\log 2}}\approx 3.3219\approx {\frac {10}{3}}}$, or ${\displaystyle 2\approx 10^{3/10}}$ to within 0.3%. This relationship is used in engineering, for example to approximate a factor of two in power as 3 dB (actual is 3.0103 dB), or to relate a kibibyte to a kilobyte.
• This coincidence can also be expressed as ${\displaystyle 128=2^{7}\approx 5^{3}=125}$ (eliminating common factor of ${\displaystyle 2^{3}}$, so also correct to 2.4%), which corresponds to the rational approximation ${\displaystyle \textstyle {\frac {\log 5}{\log 2}}\approx 2.3219\approx {\frac {7}{3}}}$, or ${\displaystyle 2\approx 5^{3/7}}$ (also to within 0.3%). This is invoked for instance in shutter speed settings on cameras, as approximations to powers of two (128, 256, 512) in the sequence of speeds 125, 250, 500, etc.

Concerning musical intervals

• The coincidence ${\displaystyle 2^{19}\approx 3^{12}}$, from ${\displaystyle {\frac {\log 3}{\log 2}}\approx 1.5849\dots \approx {\frac {19}{12}}}$ leads to the observation commonly used in music to relate the tuning of 7 semitones of equal temperament to a perfect fifth of just intonation: ${\displaystyle 2^{7/12}\approx 3/2;}$, correct to about 0.1%. The just fifth is the basis of Pythagorean tuning and most known systems of music. From the consequent approximation ${\displaystyle {(3/2)}^{12}\approx 2^{7},}$ it follows that the circle of fifths terminates seven octaves higher than the origin.
• The coincidence ${\displaystyle {\sqrt[{12}]{2}}{\sqrt[{7}]{5}}=1.33333319\ldots \approx {\frac {4}{3}}}$ leads to the rational version of 12-TET, as noted by Johann Kirnberger.
• The coincidence ${\displaystyle {\sqrt[{8}]{5}}{\sqrt[{3}]{35}}=4.00000559\ldots \approx 4}$ leads to the rational version of quarter-comma meantone temperament.
• The coincidence ${\displaystyle {\sqrt[{9}]{0.6}}{\sqrt[{28}]{4.9}}=0.99999999754\ldots \approx 1}$ leads to the very tiny interval of ${\displaystyle 2^{9}3^{-28}5^{37}7^{-18}}$ (about a millicent wide), which is the first 7-limit interval tempered out in 103169-TET.
• The coincidence of powers of 2, above, leads to the approximation that three major thirds concatenate to an octave, ${\displaystyle {(5/4)}^{3}\approx {2/1}}$. This and similar approximations in music are called dieses.

Numerical expressions

Concerning powers of π

• ${\displaystyle \pi \approx {\frac {355}{113}}=3.1415929...}$.
• ${\displaystyle \pi ^{2}\approx 10;}$ correct to about 1.3%. This can be understood in terms of the formula for the zeta function ${\displaystyle \zeta (2)=\pi ^{2}/6.}$ This coincidence was used in the design of slide rules, where the "folded" scales are folded on ${\displaystyle \pi }$ rather than ${\displaystyle {\sqrt {10}},}$ because it is a more useful number and has the effect of folding the scales in about the same place.
• ${\displaystyle \pi \approx 600/191,}$ The solid angle of the 120-cell of 4 dimensions is 191/600 of the 4-sphere, is very close to 1/pi. This is also the approximation that 1 radian is 57.3 degrees.
• ${\displaystyle \pi \approx 377/120}$ This approximation is ${\displaystyle 6F_{14}/5F_{12}}$. The polygon of 377 sides crosses a circle of diameter 120 on every edge. As 3° 8' 30" it is an approximation to pi in sexagesimal numbers.
• ${\displaystyle \pi ^{2}\approx 227/23,}$ correct to 0.0004%.
• ${\displaystyle \pi ^{2}\approx 800/81,}$ If the quadrant arc is 10, the chord is 9, is known to the ancient Egyptians. A litre represented as a cylinder of 16 inches circumference and three inches high (48 Hoppus inches) is equal to a cylinder 0.3 feet in circumference and 6 inches high, (0.045 cyl. ft) are equal with this pi. The reflective cell of the symmetry of the [3,3,5] is equal to unity, when the radius of the 4-sphere is taken as 9.
• ${\displaystyle \pi ^{3}\approx 31,}$ correct to 0.02%.
• ${\displaystyle {\sqrt {\pi }}\approx 39/22}$ to 0.015%.
• ${\displaystyle {\sqrt[{5}]{\pi ^{3}+1}}\approx 2,}$ correct to 0.004%.
• ${\displaystyle \pi \approx \left(9^{2}+{\frac {19^{2}}{22}}\right)^{1/4},}$ or ${\displaystyle 22\pi ^{4}\approx 2143;}$ accurate to 8 decimal places (due to Ramanujan: Quarterly Journal of Mathematics, XLV, 1914, pp. 350–372). Ramanujan states that this "curious approximation" to ${\displaystyle \pi }$ was "obtained empirically" and has no connection with the theory developed in the remainder of the paper.

${\displaystyle {\sqrt[{4}]{\frac {2143}{22}}}=3.1415926525\dots }$

${\displaystyle {\sqrt[{5}]{306}}=3.14155\dots }$

${\displaystyle {\sqrt[{6}]{\frac {17305}{18}}}=3.1415924\dots }$

${\displaystyle {\sqrt[{7}]{\frac {21142}{7}}}=3.14159\dots }$

Some plausible relations hold to a high degree of accuracy, but are nevertheless coincidental. One example is

${\displaystyle \int _{0}^{\infty }\cos(2x)\prod _{n=1}^{\infty }\cos \left({\frac {x}{n}}\right)\mathrm {d} x\approx {\frac {\pi }{8}}.}$

The two sides of this expression only differ after the 42nd decimal place.

Containing both π and e

• ${\displaystyle \pi ^{4}+\pi ^{5}\approx e^{6}}$, within 0.000 005%
• ${\displaystyle \pi \approx {\frac {\ln 199148648}{\sqrt {37}}}}$ to better than 14 decimal places.
• ${\displaystyle {\sqrt[{4}]{3^{3}e^{\pi }}}}$ is very close to 5, within 0.008%.
• ${\displaystyle {3}^{\frac {\pi +e}{4}}}$ is also very close to 5, approximately 0.000 538% error (Joseph Clarke, 2015)
• ${\displaystyle e^{\pi }-\pi \approx 19.99909998}$ is very close to 20 (Conway, Sloane, Plouffe, 1988); this is equivalent to ${\displaystyle (\pi +20)^{i}=-0.9999999992\ldots -i\cdot 0.000039\ldots \approx -1}$
• ${\displaystyle \pi ^{3^{2}}/e^{2^{3}}=9.9998\ldots \approx 10}$

• ${\textstyle e^{-{\frac {\pi }{9}}}+e^{-4{\frac {\pi }{9}}}+e^{-9{\frac {\pi }{9}}}+e^{-16{\frac {\pi }{9}}}+e^{-25{\frac {\pi }{9}}}+e^{-36{\frac {\pi }{9}}}+e^{-49{\frac {\pi }{9}}}+e^{-64{\frac {\pi }{9}}}=1.00000000000105...\approx 1}$. In fact, this generalizes to the approximate identity:${\displaystyle \sum _{k=0}^{n-1}{e^{-{\frac {k^{2}\pi }{n}}}}\approx {\frac {1+{\sqrt {n}}}{2}}}$ which can be explained by the Jacobian theta functional identity.
• 137 ≈ ${\displaystyle \alpha }$ * [137 ${\displaystyle \pi ^{3}}$* ${\displaystyle e}$ * φ + ${\displaystyle \pi ^{2}}$ *${\displaystyle e^{2}}$ + ${\displaystyle e^{3}}$] to within about 0.0123%, where ${\displaystyle \alpha }$ = alpha fine struct = 0.0072973525664(17) and φ = Golden Ratio to 74 decimal places (1.618...).

Containing π or e and 163

• ${\displaystyle {163}\cdot (\pi -e)\approx 69}$, within 0.0005%
• ${\displaystyle {\frac {163}{\ln 163}}\approx 2^{5}}$, within 0.000004%
• Ramanujan's constant: ${\displaystyle e^{\pi {\sqrt {163}}}\approx (2^{6}\cdot 10005)^{3}+744}$, within ${\displaystyle 2.9\cdot 10^{-28}\%}$, discovered in 1859 by Charles Hermite. This very close approximation is not a typical sort of accidental mathematical coincidence, where no mathematical explanation is known or expected to exist (as is the case for most others here). It is a consequence of the fact that 163 is a Heegner number.

Concerning logarithms

• ${\displaystyle \ln 2\approx \left({\frac {2}{5}}\right)^{\frac {2}{5}}}$, correct to 0.00024%
• ${\displaystyle \log _{9}30\approx \log _{8}25}$, within 0.000028%. Equivalently: ${\displaystyle \log {2}+\log {3}+\log {5}\approx {\frac {4}{3}}{\frac {\log {3}\log {5}}{\log {2}}}}$ (in all bases).
• ${\displaystyle \ln 2\approx {\sqrt {\ln \varphi }}}$, correct to 0.08%
• ${\displaystyle \ln x+\log _{10}x\approx \log _{2}x}$, correct to 0.6% (for all ${\displaystyle x>0}$).
• ${\displaystyle \ln 17\approx 17/6}$ to 0.004%
• ${\displaystyle 71\ln 7\approx 32\ln 75}$ to 0.000 001 %
• ${\displaystyle 6^{\phi }\approx \phi ^{6}\approx 18}$ Both of these show up nicely in base 18. In the first, the fibonacci numbers are ${\displaystyle F_{n+6}=18F_{n}-F_{n-6}}$ while the multiplication series ${\displaystyle PF_{n+1}=PF_{n}\times PF_{n-1}}$ starting with 2, 3 runs through 18, and descends to 243/256 and 32/27.
• ${\displaystyle 9^{\phi }\approx 35}$ shows in the logarithm-approximation log(3) / 13 ~ log(5) / 19 ~ log(7) / 23.

Other numerical curiosities

• ${\displaystyle 10!=6!\cdot 7!=1!\cdot 3!\cdot 5!\cdot 7!}$.
• ${\displaystyle \,2^{3}=8}$ and ${\displaystyle 3^{2}=9}$ are the only non-trivial (i.e. at least square) consecutive powers of positive integers (Catalan's conjecture).
• ${\displaystyle \,4^{2}=2^{4}}$ is the only positive integer solution of ${\displaystyle a^{b}=b^{a}}$, assuming that ${\displaystyle a\neq b}$
• The Fibonacci number F₂₉₆₁₈₂ is (probably) a semiprime, since F₂₉₆₁₈₂ = F₁₄₈₀₉₁ × L₁₄₈₀₉₁ where F₁₄₈₀₉₁ (30949 digits) and the Lucas number L₁₄₈₀₉₁ (30950 digits) are simultaneously probable primes.
• ${\displaystyle 31}$, ${\displaystyle 331}$, ${\displaystyle 3331}$, ${\displaystyle 33331}$, ${\displaystyle 333331}$, ${\displaystyle 3333331}$, and ${\displaystyle 33333331}$ are all primes, but next such number is not prime: ${\displaystyle 333333331=17\cdot 19607843}$.
• In a discussion of the birthday problem, the number ${\displaystyle \lambda ={\frac {1}{365}}{23 \choose 2}={\frac {253}{365}}}$ occurs, which is "amusingly" equal to ${\displaystyle \ln(2)}$ to 4 digits.

Decimal coincidences

• ${\displaystyle 2^{5}\cdot 9^{2}=2592}$. This makes 2592 a nice Friedman number.
• ${\displaystyle 3^{3}+4^{4}+3^{3}+5^{5}=3435}$. This makes 3435 a base-10 Münchhausen number. In fact, other than 0 and 1, the number 3435 is the only base-10 Münchhausen number. However, if one is willing to, for these purposes, adopt the convention that ${\displaystyle 0^{0}=0}$, then 438579088 is another Münchhausen number. For a few months, the number 3435 was the favorite number of Matt Parker, a stand-up mathematician, but he soon "got bored" and switched his favorite number to other types of numbers, such as narcissistic numbers and perfect numbers. Parker vehemently disagrees with the notion that 438579088 is another Münchhausen number.
• ${\displaystyle \,1!+4!+5!=145}$. The only such factorions (in base 10) are 1, 2, 145, 40585.
• ${\displaystyle (25+9)+(25\cdot 9)=259}$. In fact, this works so long as the latter part of the number is one less than a power of 10. For instance, ${\displaystyle (42+9)+(42\cdot 9)=429}$ and ${\displaystyle (42+99)+(42\cdot 99)=4299}$.
• ${\displaystyle {\frac {16}{64}}={\frac {1\!\!\!\not 6}{\not 64}}={\frac {1}{4}}}$,    ${\displaystyle {\frac {26}{65}}={\frac {2\!\!\!\not 6}{\not 65}}={\frac {2}{5}}}$,    ${\displaystyle {\frac {19}{95}}={\frac {1\!\!\!\not 9}{\not 95}}={\frac {1}{5}}}$,    ${\displaystyle {\frac {49}{98}}={\frac {4\!\!\!\not 9}{\not 98}}={\frac {4}{8}}}$ (anomalous cancellation). Also, the product of these four fractions reduces to exactly 1/100.
• ${\displaystyle (8+1)^{2}=81}$, 81 is the only such number other than 0 and 1.
• ${\displaystyle \,(4+9+1+3)^{3}=4{,}913}$; ${\displaystyle \,(5+8+3+2)^{3}=5{,}832}$; and ${\displaystyle \,(1+9+6+8+3)^{3}=19{,}683}$.
• ${\displaystyle \,2^{7}-1=127}$. This can also be written ${\displaystyle \,127=-1+2^{7}}$, making 127 the smallest nice Friedman number.
• ${\displaystyle \,1^{3}+5^{3}+3^{3}=153}$ ; ${\displaystyle \,3^{3}+7^{3}+0^{3}=370}$ ; ${\displaystyle \,3^{3}+7^{3}+1^{3}=371}$ ; ${\displaystyle \,4^{3}+0^{3}+7^{3}=407}$ — all narcissistic numbers
• ${\displaystyle \,(3+4)^{3}=343}$
• ${\displaystyle \,588^{2}+2353^{2}=5882353}$ and also ${\displaystyle \,1/17=0.05882352941\ldots }$ when rounded to 8 digits is 0.05882353. Mentioned by Gilbert Labelle in ~1980. 5882353 is also prime.
• ${\displaystyle \,123456789\cdot 8=987654312}$.
• ${\displaystyle \,2646798=2^{1}+6^{2}+4^{3}+6^{4}+7^{5}+9^{6}+8^{7}}$. The largest such number is 12157692622039623539.
• ${\displaystyle \sin(666^{\circ })=\cos(6\cdot 6\cdot 6^{\circ })=-\varphi /2}$, where ${\displaystyle \varphi }$ is the golden ratio (an amusing equality with an angle expressed in degrees)
• ${\displaystyle \,\phi (666)=6\cdot 6\cdot 6}$, where ${\displaystyle \phi }$ is Euler's totient function
• ${\displaystyle 1^{3}+2^{3}+3^{3}+4^{3}+5^{3}+6\times 6\times 6+5^{3}+4^{3}+3^{3}+2^{3}+1^{3}=666}$

Numerical coincidences in numbers from the physical world

Length of one day

The fifth hyperfactorial: 5⁵ × 4⁴ × 3³ × 2² × 1¹ = 86,400,000 is the number of milliseconds in 24 hours.

Length of six weeks

The number of seconds in six weeks, or 42 days, is exactly 10! (ten factorial) seconds (as ${\displaystyle 24=4!}$ and ${\displaystyle 42=6\cdot 7}$ and ${\displaystyle 60^{2}=5\cdot 8\cdot 9\cdot 10}$). Many have recognized this coincidence in particular because of the importance of 42 in Douglas Adams' The Hitchhiker's Guide to the Galaxy.

Length of a century

${\displaystyle \pi }$ seconds is a nano-century: 100 years × 365.25 days per year (0.25 is for a leap day every fourth year) × 24 hours per day × 3600 seconds per hour = 3,155,760,000 sec ≈ π sec × 10⁹ within 0.45%.

Speed of light

The speed of light is (by definition) exactly 299,792,458 m/s, very close to 300,000,000 m/s. This is a pure coincidence, as the meter was originally defined as 1/10,000,000 of the distance between the Earth's pole and equator along the surface at sea level, and the Earth's circumference just happens to be about 2/15 of a light-second. It is also roughly equal to one foot per nanosecond (the actual number is 0.9836 ft/ns).

Earth's diameter

The polar diameter of the Earth is equal to half a billion inches, to within 0.1%.

Gravitational acceleration

While not constant but varying depending on latitude and altitude, the numerical value of the acceleration caused by Earth's gravity on the surface lies between 9.74 and 9.87, which is quite close to 10. This means that as a result of Newton's second law, the weight of a kilogram of mass on Earth's surface corresponds roughly to 10 newtons of force exerted on an object.

This is related to the aforementioned coincidence that the square of pi is close to 10. One of the early definitions of the meter was the length of a pendulum whose half swing had a period equal to one second. Since the period of the full swing of a pendulum is approximated by the equation below, algebra shows that if this definition was maintained, gravitational acceleration measured in meters per second per second would be exactly equal to ${\displaystyle \pi ^{2}}$.

${\displaystyle T\approx 2\pi {\sqrt {\frac {L}{g}}}}$

When it was discovered that the circumference of the earth was very close to 40,000,000 times this value, the meter was redefined to reflect this, as it was a more objective standard (because the gravitational acceleration varies over the surface of the Earth). This had the effect of increasing the length of the meter by less than 1%, which was within the experimental error of the time.

Another coincidence related to the gravitational acceleration g is that its value of approximately 9.8 m/s² is equal to 1.03 light-year/year², which numerical value is close to 1. This is related to the fact that g is close to 10 in SI units (m/s²), as mentioned above, combined with the fact that the number of seconds per year happens to be close to the numerical value of c/10, with c the speed of light in m/s. In fact, it has nothing to do with SI as c/g = 354 days, nearly, and 365/354 = 1.03.

Rydberg constant

The Rydberg constant, when multiplied by the speed of light and expressed as a frequency, is close to ${\displaystyle {\frac {\pi ^{2}}{3}}\times 10^{15}\ {\text{Hz}}}$:

${\displaystyle {\underline {3.2898}}41960364(17)\times 10^{15}\ {\text{Hz}}=R_{\infty }c}$

${\displaystyle {\underline {3.2898}}68133696\ldots ={\frac {\pi ^{2}}{3}}}$

Avagadro's Number

The pound-mole, being 1 pound / 1 amu, is 273.16*10^24, against the triple-point of water at 273.16 K. This is interesting in terms of the pound-Celcius system.

US customary to metric conversions

${\displaystyle {\sqrt {62}}}$ inches equals ${\displaystyle {\sqrt {62}}\cdot 2.54=19.99997999998999998999998749998249997\ldots }$ centimeters, which is about 0.0001% less than 20 centimeters. The decimal expansion contains many nines early on.

${\displaystyle \ln 5}$ kilometers equals ${\displaystyle 1.000058\ldots }$ miles, a difference of 9.4 cm (3.7 in).

As discovered by Randall Munroe, a cubic mile is close to ${\displaystyle {\frac {4}{3}}\pi }$ cubic kilometers (within 0.5%). This means that a sphere with radius n kilometers has almost exactly the same volume as a cube with sides length n miles.

The multiplier to convert from meters per second to miles per hour is ${\displaystyle \approx {\sqrt {5}}=2\varphi -1}$. Accurate to within 0.04%. As a result, a 1 gram object moving at 100 miles per hour has almost exactly 1 joule of kinetic energy.

One furlong per fortnight is close to one centimetre per minute, to within 0.2%. The first is 201,168 cm/fortnights, the second 201,600 cm/fortnights.

Fine-structure constant

The fine-structure constant ${\displaystyle \alpha }$ is close to ${\displaystyle {\frac {1}{137}}}$ and was once conjectured to be precisely ${\displaystyle {\frac {1}{137}}}$.

${\displaystyle \alpha ={\frac {1}{137.035999074\dots }}}$

Although this coincidence is not as strong as some of the others in this section, it is notable that ${\displaystyle \alpha }$ is a dimensionless constant, so this coincidence is not an artifact of the system of units being used.