Four positive integers $a,b,c,d>1$ satisfy $\log a \log b = \log c \log d$. Is necessarily $\frac{\log a}{\log c} \in \mathbb{Q}$ or $\frac{\log a}{\log d} \in \mathbb{Q}$?

I tried to use Gelfond–Schneider theorem and Lindemann–Weierstrass theorem from transcendence theory but it didn't work.


Yes, but I can only do it with something heavier, namely Schanuel's conjecture, or more precisely the following corollary of it.

Theorem (conditional on Schanuel's conjecture): Let $p_1, p_2, \dots$ be the primes. Then $\log p_1, \log p_2, \dots$ are algebraically independent over $\mathbb{Q}$.

Proof. By unique prime factorization, $\log p_1, \log p_2, \dots$ are linearly independent over $\mathbb{Q}$. Now apply Schanuel's conjecture with $z_i = \log p_i$ for all $n$. $\Box$

If $a$ is any positive integer, then expanding out $\log a$ using the prime factorization of $a$ allows us to write it as a homogeneous linear polynomial in the variables $\log p_i$ (with rational coefficients). Hence the identity

$$\log a \log b = \log c \log d$$

asserts that some homogeneous quadratic polynomial in the variables $\log p_i$ (with rational coefficients) has two factorizations into linear factors, which must necessarily agree up to a permutation and scalar multiplication because the ring $\mathbb{Q}[\log p_i]$, being a polynomial ring, is a UFD.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.