### Numerical Coincidences

A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising.

Cosine of 355 radians is almost exactly equal to -1. Is this a coincidence? Read on!

Simple Examples

A simple example is the near-equality between 2 cubed and 3 squared. It was long ago noticed that ${2^3 = 8 \approx 9 = 3^2}$. The Belgian mathematician Eugène Charles Catalan (1814–1894) conjectured in 1844 that this was the only case of consecutive powers differing by one, in fact, the only solution in natural numbers of the diophantine equation

$\displaystyle m^p - n^q = 1$

for m, n, p and q natural numbers with p and q greater than 1. The case ${3^2 - 2^3 = 1}$ remained the only known coincidence of this kind. Catalan’s conjecture was proven in 2002 by Romanian mathematician Preda Mihailescu, so it has become Mihailescu’s theorem.

Approximations like this can be very useful. As an example, we note that ${2^{10} = 1024 \approx 1000 = 10^3}$. This is useful in estimating the size of high powers of 2:

$\displaystyle 2^{100} = (2^{10})^{10} \approx (10^3)^{10} = 10^{30}$

In elementary mechanics, it is often useful to recall that ${\pi^2 \approx 10}$. Also, that the acceleration due to gravity at the Earth’s surface is (in SI units) ${g \approx \pi^2}$, so that the period of a pendulum is ${2 \pi\sqrt{\ell/g} \approx 2 \sqrt{\ell}}$ where ${\ell}$ is the length of the pendulum. This gives a period of 2 seconds for a pendulum of length 1 metre.

Numerical Near Misses

There are a number of surprising quantities that are “almost integers”. One example is

$\displaystyle e^\pi - \pi = 19.999\,099\,98\dots \approx 20 - 10^{-4}$

There is no known reason why Gelfond’s constant ${e^\pi}$ is so close to ${\pi+20}$.

Another example concerns the sine of an angle involving this year (2017):

$\sin ( 2017 \times \sqrt[5]{2} ) = -0.999\,999\,999\,999\,999\,978\dots \approx -1 + 2.15 \times 10^{-17}$

Another near miss involves powers of ${e}$ and of ${\pi}$ and is:

$\displaystyle \pi^9 \approx 10\,e^8 \qquad\mbox{or specifically}\qquad 10 - \frac{\pi^9}{e^8} = 0.000\,161 \,2\dots$

There are many other examples like this (see Mathworld, “Almost Integers”).

The powers of the golden number ${\phi = (1+\sqrt{5})/2}$ approach integers rapidly as the power index increases. Thus

$\displaystyle \begin{array}{rcl} \phi^{30} &=& 1\,860\,497.999\,999\,5\dots \\ \phi^{60} &=& 3\,461\,452\,808\,001.999\,999\,999\,999\,7\dots \\ \phi^{90} &=& 6\,440\,026\,026\,380\,244\,498.000\,000\,000\,00\dots \end{array}$

This is not a coincidence, but is due to the fact that ${\phi}$ is in a class of numbers known as Pisot-Vijayaraghavan or PV numbers. They have the property that their powers approach integer values at an exponential rate.

Rational approximations to ${\pi}$ give almost integer values for some trigonometric quantities. For example, since ${\pi\approx 22/7}$, we find that ${\cos(22) \approx \cos(7\pi) \approx -0.999961}$. Using a higher continued fraction approximation ${\pi\approx 355/113}$ we get closer to an integral value:

$\displaystyle \cos(355) \approx -0.999\,999\,999\,545 \quad\mbox{or}\quad \cos(355) \approx - 1 + 4.543\times 10^{-10}$

Ramanujan’s Number

One of the most remarkable close approximations is Ramanujan’s Number:

$\displaystyle R = \exp(\pi\sqrt{163}) = 262\,537\,412\,640\,768\,743\,.\,999\,999\,999\,999\,250 \dots$

This is amazingly close to a whole number, differing only in the 13th decimal place:

$\displaystyle 1 - (R-[R]) \approx 7.5 \times 10^{-13}$

This is not really a coincidence, but has a theoretical explanation. It is associated with the fact that 163 is a Heegner Number. Gauss found that for certain numbers ${d}$ the integer ring ${\mathbb{Z}[\sqrt{-d}]}$ of all complex numbers of the form ${m + n\sqrt{-d}}$, where m and n are integers, is a unique factorization domain. He conjectured that this happens only for the numbers

$\displaystyle H = \{ 1, 2, 3, 7, 11, 19, 43, 67, 163 \}$

This was proved by Kurt Heegner in 1952 and these numbers are now called Heegner numbers. The theory behind them provided understanding as to why ${\exp(\pi\sqrt{d})}$ is almost integral for ${d \in H}$. So we have:

$\displaystyle \begin{array}{rcl} \exp(\pi\sqrt{19}) &\approx& 12^3(3^2-1)^3 + 744 - 0.22\dots \\ \exp(\pi\sqrt{43}) &\approx& 12^3(9^2-1)^3 + 744 -0.000\,22\dots \\ \exp(\pi\sqrt{67}) &\approx& 12^3(21^2-1)^3 + 744 - 0.000\,001\,3\dots \\ \exp(\pi\sqrt{163}) &\approx& 12^3(231^2-1)^3 + 744 - 0.000\,000\,000\,000\,75\dots \end{array}$

Sources

${\bullet}$ Mathworld. Almost Integers

${\bullet}$ Wikipedia article Heegner Numbers

${\bullet}$ Wikipedia article Pisot-Vijayaraghavan Number