Four exponentials conjecture

While a 2×2 matrix having linearly independent rows and columns usually means it has rank 2, in this case we require linear independence over a smaller field so the rank isn't forced to be 2.

For example, the matrix has rows and columns that are linearly independent over the rational numbers, since π is irrational.

[1] A special case of the conjecture is mentioned in a 1944 paper of Leonidas Alaoglu and Paul Erdős who suggest that it had been considered by Carl Ludwig Siegel.

[2] An equivalent statement was first mentioned in print by Theodor Schneider who set it as the first of eight important, open problems in transcendental number theory in 1957.

Using Euler's identity this conjecture implies the transcendence of many numbers involving e and π.

It is this consequence, for any two primes (not just 2 and 3), that Alaoglu and Erdős desired in their paper as it would imply the conjecture that the quotient of two consecutive colossally abundant numbers is prime, extending Ramanujan's results on the quotients of consecutive superior highly composite number.

So L∗ is the set of all complex numbers of the form for some n ≥ 0, where all the βi and αi are algebraic and every branch of the logarithm is considered.

The four exponentials conjecture rules out a special case of non-trivial, homogeneous, quadratic relations between logarithms of algebraic numbers.

One case of non-homogeneous quadratic relations is covered by the still open three exponentials conjecture.

Writing q = e2πiτ for the nome and j(τ) = J(q), Daniel Bertrand conjectured that if q1 and q2 are non-zero algebraic numbers in the complex unit disc that are multiplicatively independent, then J(q1) and J(q2) are algebraically independent over the rational numbers.