# Publishing mathematical coincidences

I've always wondered how some "mathematical coincidences" are published, or spread to a wider audience. For instance, the almost integer:

$$e^\pi -\pi = 19.9990999\ldots$$

was "noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe" (Weisstein).

Were these "observations" published somewhere, and if so, what motivated such a publication, knowing that there was no explanation for the coincidence?

• To tell the truth, I don't see here anything inspiring: wolframalpha.com/input/?i=e%5E%5Cpi-%5Cpi Nov 5 '17 at 14:31
• I would like to publish in such a journal the following (that is probably well known, but I discovered it when I was in high school playing with maple). $\pi - e + \gamma = 1.0005264...$ Nov 5 '17 at 20:36
• I see a big difficulty in agreeing on the meaning of a "coincidence". Suppose I am able to prove that a certain natural mathematical calculation turned out to be equal to number represented by the sequence of digits appearing in the first five license plates I saw when driving to work this morning. I would probably be very impressed, but would you?
– Ruy
Nov 5 '17 at 22:33
• Is $1.00000631046764=\tan{\frac{\pi}{3}}-\tan{\frac{\pi}{5}}-\tan{\frac{\pi}{571}}$ almost integer? Jul 11 '18 at 16:26
• Curiously $$\exp(\pi)-\frac{42}{6887}-\log23=19.9990999\ldots$$ Here $$\frac{42}{6887}=\cfrac{1}{\color{red}{163} + \cfrac{1}{1 + \cfrac{1}{\color{red}{41}}}}=\frac{2\cdot3\cdot7}{71\cdot97}.$$ Jul 10 '20 at 11:47

You answered your question yourself: it is published on the web site that you refer to. The author of the web site cites his sources, in most cases these are personal communications. Many results of this sort are spread by correspondence, on Internet, and by oral personal communication. There is also a journal "Experimental Mathematics" which publishes these kinds of observations, even those for which the author has no explanation.

There is actually the possibility for publishable research on this topic, in the context of computational complexity: How many formulas should one try for a relative accuracy of $10^{-p}$? The answer is $O(10^{p/2})$, see Seemingly Remarkable Mathematical Coincidences Are Easy to Generate by William Press (2009).$^\ast$

How hard is it to find such coincidences? Must one try on the order of $10^{19}$ formulas before getting one as good as equation (4)? [whose error is only about $2 \times 10^{−19}$] Not at all. The standard cryptographic technique of "meet-in-the-middle attack" yields a computational complexity of order the square root of the inverse precision, requiring also about this much memory. So coincidences as good as equation (4) can readily be found in $O(10^9)$ operations on an ordinary desktop machine with $O(10^9)$ memory. The rest of this note gives the details of one such implementation.

$^\ast$ As far as I could check, this article was not published in a journal, but the point of my answer is that it could have been.

• Far from being a toy subject, this type of issue is actually of critical importance in design of cryptography, where it turns out it's easy to construct "natural looking" constants that are actually parameters for weakening or backdooring a system. Nov 5 '17 at 17:18
• @R.. I suppose this has to do with the nothing up my sleeve numbers? Nov 6 '17 at 6:20
• Yes, that was the name I was reaching for but couldn't remember. Nov 6 '17 at 16:20
• This answer seems like a misleading description of the results of the linked paper. This answer makes it sound like just trying that many formulas produces that level of accuracy, when the actual procedure requires a more sophisticated meet-in-the-middle attack. Just naively trying formulas would require closer to O(10^p) tries. Nov 6 '17 at 21:11
• @user2357112 --- point taken, I have added the description from the linked paper, to avoid any misunderstanding. Thank you for the feedback. Nov 6 '17 at 21:39

A few of those that pop up on the internet every now and then (for instance on xkcd) are actually found using RIES, a program made for finding algebraic approximations given a decimal (possibly rounded) representation, by Robert Munafo.

This makes it easy to find such coincidences and many of those have been published. But these usually fall more under recreational or experimental math, as they are not appearing in the context of some "serious" research.

It is argued in the paper http://cogprints.org/3667/ (Coincidence, data compression, and Mach’s concept of “economy of thought”, by J.S. Markovitch) that most likely it is not simply a coincidence that $e^\pi-\pi$ is an almost integer. However I don't find the arguments very convincing.

P.S. A random thought. It is known (see http://aapt.scitation.org/doi/10.1119/1.3456565 The rolling sphere, the quantum spin, and a simple view of the Landau–Zener problem, by A.G. Rojo and A.M. Bloch) that $e^{-\pi}$ is related to the rotation angle of the North pole of the unit sphere when the sphere rolls along the Cornu spiral $\varphi=\frac{1}{4}s^2$ from its one pole to the another. So, if the mentioned approximate identity is not a coincidence, there should exist some approximate description of this rolling that produces $e^{-\pi}\approx \frac{1}{20+\pi}$.

• Honestly, the paper by Markovitch contains some very poorly researched problems. He would find that the behaviour of the Golden mean (equations (3), (5)) is shared by an infinite class of algebraic integers or all degrees, namely the Pisot numbers, and actually we have that for an algebraic $x$, $d(x^n,\mathbb{Z})\to0$ if and only if $x$ is Pisot. So certainly this one is anything but a coincidence.
– yo'
Nov 7 '17 at 17:10

If a coincidence is really striking and hints at some unknown theory behind it, it's worth publishing as a regular article. This adorable $1\!\tfrac12$-page paper contains nothing but "numerology"; eventually, it has led to the monstrous moonshine theory.

Also, this question seems relevant.

• The 'adorable' paper is Thompson's Some Numerology between the Fischer‐Griess Monster and the Elliptic Modular Function, published in the Bulletin of the LMS. Jun 19 '18 at 7:22