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.
A simple example is the near-equality between 2 cubed and 3 squared. It was long ago noticed that . 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
for m, n, p and q natural numbers with p and q greater than 1. The case 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 . This is useful in estimating the size of high powers of 2:
In elementary mechanics, it is often useful to recall that . Also, that the acceleration due to gravity at the Earth’s surface is (in SI units) , so that the period of a pendulum is where 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
There is no known reason why Gelfond’s constant is so close to .
Another example concerns the sine of an angle involving this year (2017):
Another near miss involves powers of and of and is:
There are many other examples like this (see Mathworld, “Almost Integers”).
The powers of the golden number approach integers rapidly as the power index increases. Thus
This is not a coincidence, but is due to the fact that 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 give almost integer values for some trigonometric quantities. For example, since , we find that . Using a higher continued fraction approximation we get closer to an integral value:
One of the most remarkable close approximations is Ramanujan’s Number:
This is amazingly close to a whole number, differing only in the 13th decimal place:
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 the integer ring of all complex numbers of the form , where m and n are integers, is a unique factorization domain. He conjectured that this happens only for the numbers
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 is almost integral for . So we have:
Mathworld. Almost Integers
Wikipedia article Heegner Numbers
Wikipedia article Pisot-Vijayaraghavan Number