Approximate three or more numbers

Question

I know that using continued fraction I can approximate $$\frac{\ln 3}{\ln 2},$$ which means that i can find two integers such that $2^{k_2} \approx 3^{k_3}$, my question is can I approximate using continued fraction or any other method three integer $k_2 ,k_3,k_5$ such that $2^{k_2} \approx 3^{k_3} \approx 5^{k_5}$ ?

Answer 1

In general, you can find infinitely many integers that fulfill your requirement (or, perhaps, none at all, depending on what you mean by "$\approx$"). However, there are likely some constraints that you might wish to impose. To start, I'm going to rephrase your question slightly, just to make myself more comfortable.

Suppose that $\alpha_1, \alpha_2, \dotsc, \alpha_n$ are real numbers. Further suppose that they are mutually incommensurable (i.e. $\frac{a_i}{a_j} \not\in\mathbb{Q}$ for all $i\ne j$). For a fixed $\varepsilon > 0$, is there an efficient way to find $Q,p_1,p_2,\dotsc,p_n \in \mathbb{Z}$ such that

1. $\left|Q\alpha_i - p_i \right| < \varepsilon$, and
2. $Q$ is not "very big" (by choosing $Q > \varepsilon^{-1}$, I can find lots of good $p_i$—can we do better?).

That is, can I approximate each of the $\alpha_i$ with rational numbers that all have the same denominator, where that denominator is not too big?

I do not have the expertise to give a complete answer to your question, but let me offer a tentative "yes." You can first approximate each of the $\alpha_i$ via, say, continued fraction approximation. You can then find a common denominator via some kind of basis reduction algorithm, such as the LLL algorithm. This process is outlined in chapter 9 of the text by Bremner, cited below. Further references are also provided.

Bremner, Murray R., Lattice basis reduction. An introduction to the LLL algorithm and its applications, Pure and Applied Mathematics (Boca Raton) 300. Boca Raton, FL: CRC Press (ISBN 978-1-4398-0702-6/hbk; 978-1-4398-0704-0/ebook). xvii, 316 p. (2012). ZBL1237.68007.

Kaib, Michael; Schnorr, Claus P., The generalized Gauss reduction algorithm, J. Algorithms 21, No. 3, 565-578 (1996). ZBL0876.68049.

Lenstra, A.K.; Lenstra, H.W.jun.; Lovász, László, Factoring polynomials with rational coefficients, Math. Ann. 261, 515-534 (1982). ZBL0488.12001.

Answer 2

$2 = 3^{\log_3 2} = 5^{\log_5 2}$

Set $\log_3 2 = \frac {\ln 2}{\ln 3} \approx \frac pq$ and $\log_5 2 =\frac {\ln 2}{\ln 5} \approx \frac rs$ approximated as closely as you like.

$2^{qs} \approx 3^{ps} \approx 5^{rq}$.

(However the margin of error will be significantly higher. But you can make the error as small as you like but making the initial error between $\frac pq$ and $\frac rx$ and $\log_3 2$ and $\log_5 2$ exponentially smaller.)

Example $\log_3 2 \approx .63$ and $\log_5 2 \approx .43$

So $2^{100} \approx 3^{63} \approx 5^{43}$

$ 1267650600228229401496703205376\approx 1144561273430837494885949696427 \approx 1136868377216160297393798828125$