Four Exponentials Conjecture - Strong Four Exponentials Conjecture

The strongest result that has been conjectured in this circle of problems is the strong four exponentials conjecture. This result would imply both aforementioned conjectures concerning four exponentials as well as all the five and six exponentials conjectures and theorems, as illustrated to the right, and all the three exponentials conjectures detailed below. The statement of this conjecture deals with the vector space over the algebraic numbers generated by 1 and all logarithms of non-zero algebraic numbers, denoted here as L∗. So L∗ is the set of all complex numbers of the form

for some n ≥ 0, where all the β_i and α_i are algebraic and every branch of the logarithm is considered. The statement of the strong four exponentials conjecture is then as follows. Let x₁, x₂, and y₁, y₂ be two pairs of complex numbers with each pair being linearly independent over the algebraic numbers, then at least one of the four numbers x_i y_j for 1 ≤ i,j ≤ 2 is not in L∗.