What's the "most efficient" way to encode that $\ln(2):\ln(3):\ln(5):\ln(7) \approx 171:271:397:480$ using $3$ approximations?
For example:
- $((\frac{3^3}{5^2})^3)^3 \approx 2$ uses $3+2+3+3 = 11$ exponentiations and $3$ bases for a total "complexity" of $14$
- $(((\frac{2^2}{3})^2)^2)^2 \approx 2\cdot5$ uses $2+2+2+2 = 8$ exponentiations and $4$ bases for a total "complexity" of $12$
- $(7\cdot(\frac{3}{5})^2)^3\approx(2^2)^2$ uses $2+3+2+2 = 9$ exponentiations and $4$ bases for a total "complexity" of $13$
This encoding has a "complexity" of $39$. Can we prove this is optimal or does there exist a simpler encoding?
Feel free to redefine "complexity" in a better way to measure the simplicity of the approximations.
Motivation for this problem:
- the ratio corresponds to the the 171 term in https://oeis.org/A117536 which relates finding rational approximations of logarithms to the Riemann Hypothesis which has applications in musical tuning https://en.xen.wiki/w/The_Riemann_zeta_function_and_tuning
- The $3$ approximations are related to the abc conjecture with the triples $(a,b,c) =(1, 2400 ,2401), (1, 4374, 4375), (37, 32768, 32805)$ which are the 5th, 19th, and 37th highest quality triples (https://www.math.leidenuniv.nl/~desmit/abc/index.php?set=2)