Four Exponentials Conjecture - Sharp Four Exponentials Conjecture

Sharp Four Exponentials Conjecture

The four exponentials conjecture reduces the pair and triplet of complex numbers in the hypotheses of the six exponentials theorem to two pairs. It is conjectured that this is also possible with the sharp six exponentials theorem, and this is the sharp four exponentials conjecture. Specifically, this conjecture claims that if x₁, x₂, and y₁, y₂ are two pairs of complex numbers with each pair being linearly independent over the rational numbers, and if β_ij are four algebraic numbers for 1 ≤ i,j ≤ 2 such that the following four numbers are algebraic:

then x_i y_j = β_ij for 1 ≤ i,j ≤ 2. So all four exponentials are in fact 1.

This conjecture implies both the sharp six exponentials theorem, which requires a third x value, and the as yet unproven sharp five exponentials conjecture that requires a further exponential to be algebraic in its hypotheses.