Have any long-suspected irrational numbers turned out to be rational? - MathOverflow

Have any long-suspected irrational numbers turned out to be rational?

2010-07-22T16:06:17Z

The history of proving numbers irrational is full of interesting stories, from the ancient proofs for $\sqrt{2}$, to Lambert's irrationality proof for $\pi$, to Roger Apéry's surprise demonstration that $\zeta(3)$ is irrational in 1979.

There are many numbers that seem to be waiting in the wings to have their irrationality status resolved. Famous examples are $\pi+e$, $2^e$, $\pi^{\sqrt 2}$, and the Euler–Mascheroni constant $\gamma$. Correct me if I'm wrong, but wouldn't most mathematicians find it a great deal more surprising if any of these numbers turned out to be rational rather than irrational?

Are there examples of numbers that, while their status was unknown, were "assumed" to be irrational, but eventually shown to be rational?

Answer by Gjergji Zaimi for Have any long-suspected irrational numbers turned out to be rational?

2010-07-22T16:25:32Z

I don't think Legendre expected this number to be rational, let alone integer...

Answer by Marty for Have any long-suspected irrational numbers turned out to be rational?

2010-07-22T16:58:26Z

It's hazardous to guess the reactions of most mathematicians. But I imagine a very large number of mathematicians would be surprised if Schanuel's conjecture turned out to be false. And this conjecture implies the irrationality of 3/4 of your examples, I think.

As for the Euler–Mascheroni constant, I have never thought about it.

Answer by Greg Kuperberg for Have any long-suspected irrational numbers turned out to be rational?

2010-07-22T20:39:52Z

There are reasons that any modern example is likely to resemble the status of Legendre's constant. Most (but not all) interesting numbers admit a polynomial-time algorithm to compute their digits. In fact, there is an interesting semi-review by Borwein and Borwein that shows that most of the usual numbers in calculus (for example, $\exp(\sqrt{2}+\pi)$) have a quasilinear time algorithm on a RAM machine, meaning $\tilde{O}(n) = O(n(\log n)^\alpha)$ time to compute $n$ digits. Once you have $n$ digits, you can use the continued fraction algorithm to find the best rational approximation with at most $n/2-O(1)$ digits in the denominator. The continued fraction algorithm is equivalent to the Euclidean algorithm, which also has a quasilinear time version according to Wikipedia.

Euler's constant has been to computed almost 30 billion digits, using a quasilinear time algorithm due to Brent and McMillan.

As a result, for any such number it's difficult to be surprised. You would need a mathematical coincidence that the number is rational, but with a denominator that is out of reach for modern computers. (This was Brent and MacMillian's stated motivation in the case of Euler's constant.) I think that it would be fairly newsworthy if it happened. On the other hand, if you can only compute the digits very slowly, then your situation resembles Legendre's.

I got e-mail asking for a reference to the paper of Borwein and Borwein. The paper is On the complexity of familiar functions and numbers. To summarize the relevant part of this survey paper, any value or inverse value of an elementary function in the sense of calculus, including also hypergeometric functions as primitives, can be computed in quasilinear time. So can the gamma or zeta function evaluated at a rational number.

Answer by paul Monsky for Have any long-suspected irrational numbers turned out to be rational?

2010-07-22T21:50:23Z

Consider the hypergeometric function ${}_2F_1(a,b,c;z)$. When $a$, $b$ and $c$ are rational and ${}_2F_1$ is a transcendental function, Siegel sought to prove that--apart from obvious exceptions--the function takes transcendental values at algebraic $z$. But it turns out that there are $a$, $b$ and $c$ for which this is false. For example:

$${}_2F_1(1/3,2/3,5/6;27/32)=8/5$$

$${}_2F_1(1/4,1/2,3/4;80/81)=9/5$$

$${}_2F_1(1/12,5/12,1/2;1323/1331)= 11^{1/4}$$

Answer by Wadim Zudilin for Have any long-suspected irrational numbers turned out to be rational?

2010-07-23T00:09:40Z

Hmmm, I am upset to be not not in time for the question (a short night sleep was necessary!).

Let me comment on a quite opposite to the question

Are there examples of numbers that, while their status was unknown, were "assumed" to be irrational, but eventually shown to be rational?

There is one famous constant, the One-Ninth Constant, which for a very long time was expected to be a rational number, namely, $1/9$. It was only in the 1980s when A. Gonchar and E. Rakhmanov found an explicit formula for it through the elliptic integrals and Nesterenko's 1996 theorem on the algebraic independence of modular functions resulted in the transcendence of this constant. There is a nice chapter on this constant in Steven Finch's Mathematical Constants, Cambridge University Press 2003 (§4.5, pp. 259--262), although the transcendence is not mentioned there.

Answer by Wadim Zudilin for Have any long-suspected irrational numbers turned out to be rational?

2010-07-23T00:36:04Z

It was not known for a long time that the number $$ \frac{\zeta(2)}{\pi^2} =\frac1{\pi^2}\sum_{n=1}\frac1{n^2} $$ is rational, $1/6$. Euler showed this in his solution of the Basel problem. Related examples include $\zeta(2)^2/\zeta(4)$ and, more generally, $\zeta(2k)/\pi^{2k}$ for integer $k$. I mention this historical fact because of several attempts on MO to find a "closed form" evaluation of $\zeta(3)$ (mostly of the form $\zeta(3)/\pi^3\overset?\in\mathbb Q$, which is numerically confirmed to be doubtfully true).

EDIT. I do understand that not everybody feels this post to be in a (magic) "spirit" of the OP. But I do not understand your downvotes here. Why don't you downvote when somebody puts a problem on finding a "closed form" for $\zeta(3)$? Or when somebody "proves" that $\log2$ is a rational multiple of $\pi^2$? Anyway, I do not remove this post but put it in the community wiki mode, as it might be used, together with this answer and comments therein, as a reference to later silly questions about zeta values.

Answer by Cam McLeman for Have any long-suspected irrational numbers turned out to be rational?

2010-07-23T04:39:08Z

This certainly doesn't answer the question, but I can't help but mention Conway's constant:

http://mathworld.wolfram.com/ConwaysConstant.html

It relates to Pete's comment about "bumping" it up a notch, in that it gives an example of a number that I think any reasonable person would conjecture to be transcendental, but turns out to be algebraic (of degree 71, of all things). And algebraic numbers are sort of finitely far from being rational, so...

Answer by dke for Have any long-suspected irrational numbers turned out to be rational?

2010-08-03T13:31:23Z

Another 'opposite' example - a naturally occurring number suspected to be rational but turning out to be irrational - occurs in the study of random polytopes. In 1923, Blaschke asked

What is the expected volume of a tetrahedron with vertices chosen randomly in a unit volume tetrahedron ?

The corresponding answer for a unit line is $\frac{1}{3}$ and for a unit triangle it's $\frac{1}{12}$. Klee made the (very plausible) conjecture that for the tetrahedron the answer is $\frac{1}{60}$ but later Monte Carlo experiments suggested the answer was closer to $\frac{1}{57}$.

Then in 2001, Buchta and Reitzner showed that the answer is actually

$$\frac{13}{720}-\frac{\pi^2}{15015}.$$

Answer by Richard Stanley for Have any long-suspected irrational numbers turned out to be rational?

2010-09-25T17:49:58Z

A surprising rational number is 32/27. Thomassen showed in 1997 that the closure of the set of all real zeros of all chromatic polynomials of graphs is $\lbrace 0\rbrace \cup \lbrace 1\rbrace \cup [32/27,\infty)$.

Answer by Harrison Brown for Have any long-suspected irrational numbers turned out to be rational?

2010-09-29T05:24:23Z

Bernstein's constant doesn't strictly fit the parameters of the question, but it's notable as a more recent "Legendre-type" example. Bernstein conjectured that his constant was exactly $\frac{1}{2\sqrt{\pi}}$ in 1914; it wasn't until the '80s that it became possible to compute enough digits to refu(dia)te the conjecture.

Although perhaps it wasn't surprising -- I have no idea whether Bernstein's conjecture was generally believed; can anyone shed light on that?

Answer by Fabio for Have any long-suspected irrational numbers turned out to be rational?

2010-12-08T18:55:51Z

Dear participants (and curious), I'd like to bring your attention to two brand new arXiv papers with theorems closely related to the original question posed by I. J. Kennedy. The papers are: 1) "An interface between physics and number theory", available at http://arxiv.org/abs/1011.0523v1, and 2) "New Properties of Fourier Series and Riemann Zeta Function", available at http://arxiv.org/abs/1008.5046v3 . In the first paper, it is shown (with some controversy) that Euler's gamma is a... RATIONAL number!!! The other paper contains an interesting theorem (#22) in which the author shows that the ratio $\zeta(2n+1)/{\pi^{2n+1}}$ does not have an Euler sum. It seems (and I'm now trying to proof) that this implies these ratios are irrational. It would be interesting to hear the opinion of some specialists (number theorists) -- such as W. Zudilin and Rivoal -- on the validity of these theorems and my conclusion on the irrationality of the above ratios.

(F. M. S. Lima, University of Brasilia -- fabio{at}fis.unb.br)

Answer by GH for Have any long-suspected irrational numbers turned out to be rational?

2011-03-01T06:37:14Z

Here is another example of a number that was thought to be rational until it was proved to be irrational. Erdős conjectured that not much more integers are representable as a sum of two squareful numbers than as a sum of two squares. More precisely, he conjectured that up to $x$ the number of integers in the first set is $x/(\log x)^{1/2+o(1)}$. Blomer proved that the exponent $1/2$ is wrong, the correct value is $1-2^{-1/3}$. He also showed that the same estimate is valid for sums of a square and a squareful number. See J. London Math. Soc. (2) 71 (2005), 69-84.