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I was surprised that the numbers $\pi$, $\ln{(2)}$, $\zeta{(2)}$, and $\zeta{(3)}$ can be shown to be irrational in what seems to be "three-lined proofs" (as identified here on Overflow: Establishing zeta(3) as a definite integral and its computation.).

The idea goes that we suppose, for $j \in \mathbb{N}$, we have a family of non-zero integrals such that $$\int_0^1 x^j f(x)\ dx = P_j \xi+ Q_j $$ where $P_j, Q_j \in \mathbb{Q}$, $f(x)$ is arbitrary, and $\xi$ is some number. If $\xi$ is rational, then $\int_0^1 x^j f(x)\ dx = A_j/B_j $ yields a rational expression. Multipying $a_{nj} \in \mathbb{Z}$ to both sides, and taking the sum from $j = 0$ to $n$, we have $$\sum_{j=0}^n a_{nj} \int_0^1 x^j f(x)\ dx = \sum_{j=0}^n a_{nj} \frac{A_j}{B_j} = \frac{E_n}{F_n} $$ where $E_n, F_n \in \mathbb{Z}$.

Beuker treated the $a_{nj}$ coefficients as the coefficients of the shifted Legendre polynomials in his proof for the irrationality of $\zeta{(3)}$, where

$$ P_n{(x)} = \frac{1}{n!} \frac{d^n}{dx^n} (x^n (1-x)^n) = \sum_{j=0}^n a_{nj} \, x^j$$

This leads to

$$\int_0^1 \bigg( \sum_{j=0}^n a_{nj} \, x^j \bigg) f(x)\ dx = \int_0^1 P_n{(x)}\,f(x) \ dx = \frac{E_n}{F_n}$$

If we can show, as $n$ gets larger,

$$ \left| F_n \int_0^1 \frac{1}{n!} (x^n (1-x)^n)\, \frac{d^n}{dx^n}f(x) \ dx \right| = |E_n|\to 0 $$

we get a contradiction, since at "some point", $|E_n|$ is an integer between $0$ and $1$. Thus, $\xi$ must be irrational. Of course, the hard part is finding a suitable function $f(x)$.

Since $f(x)$ has been found for the proofs of the above numbers, has it been shown the number $e$ can be proved irrational using the above method (implying has such an $f(x)$ been found)?

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One has $$\int_0^1 x^k e^x dx = A_ke+B_k \to 0$$ as $k\to \infty$, $A_k,B_k\in \mathbb{Z}$, by integration-by-parts and induction. If $e=a/b$, then $(A_k a+B_kb)/b \to 0$, so $A_ke+B_k=0$ for $k$ large, a contradiction.

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  • $\begingroup$ Oh wow! I didn't see that one coming! Do you know of any existing literature on finding $f(x)$ to show a given $\xi$ is irrational? $\endgroup$ – Brian Diaz Apr 14 '15 at 4:30
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    $\begingroup$ Incidentally, $\int_0^1 (1-x)^k e^x dx = k!(e - \sum_{0 \le i \le k} 1/i!)$, so if you use $(1-x)^k$ instead of $x^k$ you get the usual approximation to $e$ through Taylor series. $\endgroup$ – Henry Cohn Apr 14 '15 at 4:49
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    $\begingroup$ @Henry Cohn: Yes, it's just a repackaging of the standard Fourier proof in a form reminiscent of the question. $\endgroup$ – Ian Agol Apr 14 '15 at 4:56
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    $\begingroup$ As it turns out, Beuker's already observed the irrationality of $e^a$, where $a \in \mathbb Q$, albeit the idea, as @Ian Agol mentions, dates back to Fourier. For those interested, you can find the link here. $\endgroup$ – Brian Diaz Apr 14 '15 at 16:20

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