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Motivation:

Many interesting irrational numbers (or numbers believed to be irrational) appear as answers to natural questions in mathematics. Famous examples are $e$, $\pi$, $\log 2$, $\zeta(3)$ etc. Many more such numbers are described for example in the wonderful book "Mathematical Constants" by Steven R. Finch.

The question:

I am interested in theorems where a "special" rational number makes a surprising appearance as an answer to a natural question. By a special rational number I mean one with a large denominator (and preferably also a large numerator, to rule out numbers which are simply the reciprocals of large integers, but I'll consider exceptions to this rule). Please provide examples.

For illustration, here are a couple of nice examples I'm aware of:

  1. The average geodesic distance between two random points of the Sierpinski gasket of unit side lengths is equal to $\frac{466}{885}$. This is also equivalent to a natural discrete math fact about the analysis of algorithms, namely that the average number of moves in the Tower of Hanoi game with $n$ disks connecting a randomly chosen initial state to a randomly chosen terminal state with a shortest number of moves, is asymptotically equal to $\frac{466}{885}\times 2^n$. See here and here for more information.

  2. The answer to the title question of the recent paper ""The density of primes dividing a term in the Somos-5 sequence" by Davis, Kotsonis and Rouse is $\frac{5087}{10752}$.

Rules:

1) I won't try to define how large the denominator and numerator need to be to for the rational number to qualify as "special". A good answer will maximize the ratio of the number's information theoretic content to the information theoretic content of the statement of the question it answers. (E.g., a number like 34/57 may qualify if the question it answers is simple enough.) Really simple fractions like $3/4$, $22/7$ obviously do not qualify.

2) The question the number answers needs to be natural. Again, it's impossible to define what this means, but avoid answers in the style of "what is the rational number with smallest denominator solving the Diophantine equation [some arbitrary-sounding, unmotivated equation]".

Edit: a lot of great answers so far, thanks everyone. To clarify my question a bit, while all the answers posted so far represent very beautiful mathematics and some (like Richard Stanley's and Max Alekseyev's answers) are truly astonishing, my favorite type of answers involve questions that are conceptual in nature (e.g., longest increasing subsequences, tower of Hanoi, Markov spectrum, critical exponents in percolation) rather than purely computational (e.g., compute some integral or infinite series) and to which the answer is an exotic rational number. (Note that someone edited my original question changing "exotic" to "special"; that is fine, but "exotic" does a better job of signaling that numbers like 1/4 and 2 are not really what I had in mind. That is, 2 is indeed quite a special number, but I doubt anyone would consider it exotic.)

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    $\begingroup$ If one is allowed to cheat and use quadratic irrationalities, then there is Freiman's constant: the last gap in the Markov spectrum, namely, $\frac{2221564096 + 283748\sqrt{462}}{491993569}$. $\endgroup$
    – Richard Stanley
    Commented Sep 9, 2015 at 17:12
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    $\begingroup$ Richard, that is indeed a beautiful example and very much in the spirit of what I had in mind. $\endgroup$
    – Dan Romik
    Commented Sep 9, 2015 at 18:04
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    $\begingroup$ There are two kinds of "famous" rational constants. First kind is when it is clear a priori that it is rational (like Bernoulli numbers). Second kind where this comes as a suprise. Like the examples in my answer. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 9, 2015 at 22:01
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    $\begingroup$ I don't want to type it in, but there's the simplest rational right triangle with area 157, which Koblitz tells me was computed by Zagier. $\endgroup$
    – Kimball
    Commented Sep 12, 2015 at 17:11
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    $\begingroup$ $\frac{1}{12}$ seems to make a lot of unusual appearances. It’s negative is the sum of natural numbers. It appears in coefficients of some index theorems and many other places. There might be a hitchhiker style joke you can make replacing $42$ with $\frac{1}{12}$ $\endgroup$
    – Sidharth Ghoshal
    Commented Nov 16, 2023 at 5:09

32 Answers 32

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I can hardly ever be more surprised than with this expression (by Ramanujan):

$$ \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{ (4k)! (1103+26390k) }{ (k!)^4 396^{4k} } = \frac1{\pi}. $$

In particular, it's intriguing that such "special" rational numbers (in the sense defined by the OP) appear in a series related to good old $\pi$.

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    $\begingroup$ That isn't rational. $\endgroup$
    – Christopher King
    Commented Sep 12, 2015 at 1:12
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    $\begingroup$ @PyRulez, the LHS has some cool fractions, which are rational except for the $\sqrt{2}$ factor. Regardless, it is never really wrong to cite Ramanujan... $\endgroup$
    – Dan Romik
    Commented Sep 12, 2015 at 7:36
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    $\begingroup$ To me, the "rules of the game" of the actual question are not so clear if PyRulez's objection is dismissed. $\endgroup$
    – Todd Trimble
    Commented Sep 13, 2015 at 5:24
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    $\begingroup$ @LuisMendo I can't see how this is responsive to my comment. Is the question now allowing not just single rational values, but sequences of rational values that converge to... whatever? Couldn't you then just stick a pin in Ramanujan's notebooks and wherever you land, there you have it? $\endgroup$
    – Todd Trimble
    Commented Sep 13, 2015 at 15:32
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    $\begingroup$ @DanRomik Okay, fair enough. I do note however that Richard Stanley demurred posting his response as an answer, for reasons that I think are related to the point I was trying to make. :-) $\endgroup$
    – Todd Trimble
    Commented Sep 13, 2015 at 16:46
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Interesting rational constant concerning base-10 normal numbers.

Let $a$ be a real number with a base-10 decimal representation $a_1a_2\ldots a_n \ldots$ Denote the number of ways to write $a_n$ as the sum of positive integers as $p(a_n)$ - also called the partition of $a_n$. I write $A(n)=\sum_{k=1}^n p(a_k)$ and $B(n)= \sum_{k=1}^n a_k$ and set $$\beta(a)=\lim_{n\to \infty}{A(n)\above 1.5 pt B(n)}$$ If $a$ is a base-10 normal number then $\beta(a)={97 \above 1.5 pt 45}$ .

Note the converse of the above statement is false!

Numerically ${97\above 1.5pt 45}$ can be written $2.15\ldots$. If we believe $\pi$ is normal then we would expect $\beta(\pi)={97 \above 1.5 pt 45}$. Up to the $500\text{ }000$-th digit $\beta(\pi)=2.153781\ldots$ See the plot of $\beta(\pi)$ below. Note the blue line is the constant value ${97\above 1.5 pt 45}$ and the orange "graph" are the values of $\beta(\pi)$.

enter image description here

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    $\begingroup$ As written the sum is infinite. If you fix it (to what's written in the stack exchange post) I would argue that this is unsurprising. You could replace $p$ with any rational-valued function and it would still be true with constant $(p(0) + p(1) + \cdots + p(9))/(0 + 1 + \cdots + 9)$. $\endgroup$
    – Tim Carson
    Commented Aug 22, 2017 at 19:05
  • $\begingroup$ @TimCarson I finally understand what you mean by "any rational-valued function. " I understand now. $\endgroup$
    – Anthony
    Commented Jun 26, 2018 at 19:22
  • $\begingroup$ Actually now I don't remember why I said rational-valued. It seems like any function $p : \{0, 1, ..., 9\} \to \mathbb{R}$ would do. $\endgroup$
    – Tim Carson
    Commented Jul 13, 2018 at 14:48
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