Mathematical coincidence

This article is about numerical curiosities. For the technical mathematical concept of coincidence, see coincidence point.

In mathematics, a mathematical coincidence can be said to occur when two expressions show a near-equality that lacks direct theoretical explanation.

Introduction

A mathematical coincidence often comprises an integer, and the surprising (or "coincidental") feature is the fact that a real number is close to a small integer; or, more generally, to a rational number with a small denominator.

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 esthetics could be invoked to adjudicate the value of a mathematical coincidence, and there are in fact exceptional cases of true mathematical significance (see Ramanujan's constant below, which made it into print some years ago as a scientific April Fools' joke). 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 pi

• 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, ...].
• Assuming 365.25 days per year, the number of seconds per year is approximately $\pi \times 10^{7}$ (correct to about 0.45%) and is also approximately ${\sqrt {10}}\times 10^{7}$ (correct to about 0.21%).

Concerning base 2

• The coincidence $2^{10}=1024\approx 1000=10^{3}$, correct to 2.4%, relates to the rational approximation $\textstyle {\frac {\log 10}{\log 2}}\approx 3.3219\approx {\frac {10}{3}}$, or $2\approx 10^{3/10}$ 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 kilobyte to a kibibyte; see binary prefix.

Concerning musical intervals

• The coincidence $2^{19}\approx 3^{12}$, from ${\frac {\log 3}{\log 2}}\approx 1.5849...\approx {\frac {19}{12}}$ 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: $2^{7/12}\approx 3/2;$, correct to about 0.1%. The just fifth is the basis of Pythagorean tuning and most known systems of music. From the consequent approximation ${(3/2)}^{12}\approx 2^{7},$ it follows that the circle of fifths terminates seven octaves higher than the origin.

• The coincidence ${\sqrt[{12}]{2}}{\sqrt[{7}]{5}}\approx 1.33333319...\approx {\frac {4}{3}}$ leads to the rational version of 12-TET, as noted by Johann Kirnberger.

Concerning camera settings

• The coincidence $5^{3}\approx 2^{7}$ is invoked in typical shutter speed settings on cameras, as approximations to powers of two in the sequence of speeds 125, 250, 500, etc.

Numeric expressions

Concerning powers of pi

• $\pi ^{2}\approx 10;$ correct to about 1.3%. This can be understood in terms of the formula for the zeta function $\zeta (2)=\pi ^{2}/6.$ This coincidence was used in the design of slide rules, where the "folded" scales are folded on $\pi$ rather than ${\sqrt {10}},$ because it is a more useful number and has the effect of folding the scales in about the same place.
• $\pi ^{2}\approx 227/23,$ correct to 0.0004%.
• $\pi \approx \left(9^{2}+{\frac {19^{2}}{22}}\right)^{1/4},$ or $\pi ^{4}\approx 2143/22;$ accurate to 8 decimal places (due to Ramanujan: Quarterly Journal of Mathematics, XLV, 1914, pp350-372). Ramanujan states that this "curious approximation" to $\pi$ was "obtained empirically" and has no connection with the theory developed in the remainder of the paper.

Containing both pi and e

• $\pi ^{4}+\pi ^{5}\approx e^{6}$, within 0.000 005%

• $e^{\pi }-\pi \approx 19.99909998$ is very close to 20 (Conway, Sloane, Plouffe, 1988); this is equivalent to $(\pi +20)^{i}=-0.9999999992...-i\cdot 0.000039...\approx -1$

• $\pi ^{3^{2}}/e^{2^{3}}\approx 9.9998$

Containing pi or e and number 163

• ${163}\cdot (\pi -e)\approx 69$, within 0.0005%

• Ramanujan's constant: $e^{\pi {\sqrt {163}}}\approx (2^{6}\cdot 10005)^{3}+744$, within $2.9\cdot 10^{-28}\%$. This fact was used in an April Fools' article in the April 1975 issue of Scientific American by Martin Gardner, in which it was claimed that $e^{\pi {\sqrt {163}}}$ is exactly equal to 262,537,412,640,768,744. This fact also implies $\pi \approx {1 \over {\sqrt {163}}}\log _{e}{(640320^{3}+744)}$, which is correct to about 20 digits. Similarly,

$\left[{\frac {1}{\pi }}\log _{e}(640320^{3}+744)\right]^{2}=163.00000000000000000000000000000000232\ldots$,

which I. J. Good has said is possibly "the most surprising possible approximate integer." (See also Heegner number.)

Coincidences of units

• One mile is 1.609344 km, very close to $\varphi$ (correct to about 0.5%), where $\varphi ={1+{\sqrt {5}} \over 2}\approx 1.618$ is the golden ratio; the ratio of Fibonacci numbers 8 and 5, 1.60, is another good approximation.

Other numeric curiosities

• $10!=6!\cdot 7!=1!\cdot 3!\cdot 5!\cdot 7!$.
• $\sin(666^{\circ })=\cos(6\cdot 6\cdot 6^{\circ })=-\phi /2$, where $\phi$ is the golden ratio (an amusing equality with an angle expressed in degrees) (see Number of the Beast)
• $\,\phi (666)=6\cdot 6\cdot 6$, where $\phi$ is Euler's totient function
• $\,2^{3}=8$ and $3^{2}=9\,$ are the only non-trivial (i.e. at least square) consecutive powers of positive integers (Catalan's conjecture).
• $\,4^{2}=2^{4}$ is the only positive integer solution of $a^{b}=b^{a},a\neq b$ (see Lambert's W function for a formal solution method)
• 31, 331, 3331 etc. up to 33333331 are all prime numbers, but then 333333331 is not. See also Formula for primes.
• 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.
• In a discussion of the birthday problem, the number $\lambda ={\frac {1}{365}}{23 \choose 2}$ occurs, which is "amusingly" equal to $\ln(2)$ to 4 digits.

Decimal coincidences

• $2^{5}\cdot 9^{2}=2592$. This makes 2592 a nice Friedman number.
• $\,1!+4!+5!=145$. The only such factorions are 1, 2, 145, 40585.
• ${\frac {16}{64}}={\frac {1\!\!\!\not 6}{\not 64}}={\frac {1}{4}}$,    ${\frac {26}{65}}={\frac {2\!\!\!\not 6}{\not 65}}={\frac {2}{5}}$,    ${\frac {19}{95}}={\frac {1\!\!\!\not 9}{\not 95}}={\frac {1}{5}}$,    ${\frac {49}{98}}={\frac {4\!\!\!\not 9}{\not 98}}={\frac {4}{8}}$ (anomalous cancellation)
• $\,(4+9+1+3)^{3}=4{,}913$ and $\,(1+9+6+8+3)^{3}=19{,}683$.
• $\,2^{7}-1=127$. This can also be written $\,127=-1+2^{7}$, making 127 the smallest nice Friedman number.
• $\,1^{3}+5^{3}+3^{3}=153$ ; $\,3^{3}+7^{3}+0^{3}=370$ ; $\,3^{3}+7^{3}+1^{3}=371$ ; $\,4^{3}+0^{3}+7^{3}=407$ — all narcissistic numbers
• $\,(3+4)^{3}=343$
• $\,588^{2}+2353^{2}=5882353$ and also $\,1/17=0.0588235294117647...$ when rounded to 8 digits is 0.05882353 mentioned by Gilbert Labelle in ~1980.
• $\,2646798=2^{1}+6^{2}+4^{3}+6^{4}+7^{5}+9^{6}+8^{7}$. The largest such number is 12157692622039623539.
• $1782^{12}+1841^{12}\approx 1922^{12}$ is an example of three numbers which are very close to disproving Fermat's Last Theorem for n=12, the two sides differ by about $3\times 10^{-8}\%$. The left hand side is in fact approximately equal to $1921.9999999559^{12}$ (accurate to 10 decimal places).

See also

• For a list of coincidences in physics, see anthropic principle.
• Almost integer
• Birthday problem
• Exceptional isomorphism
• Narcissistic number
• The Strong Law of Small Numbers

External links

• Template:Ru icon В. Левшин. - Магистр рассеянных наук. - Москва, Детская Литература 1970, 256 с.
• Hardy, G. H. - A Mathematician's Apology. - New York: Cambridge University Press, 1993, (ISBN 0-521-42706-1)
• Sequence OEIS: A032799 in the On-Line Encyclopedia of Integer Sequences
• Almost Integer from Wolfram MathWorld
• Various mathematical coincidences in the "Science & Math" section of futilitycloset.com