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 βi and αi 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 xi yj for 1 ≤ i,j ≤ 2 is not in L∗.
Read more about this topic: Four Exponentials Conjecture
Famous quotes containing the words strong and/or conjecture:
“A two-year-old can be taught to curb his aggressions completely if the parents employ strong enough methods, but the achievement of such control at an early age may be bought at a price which few parents today would be willing to pay. The slow education for control demands much more parental time and patience at the beginning, but the child who learns control in this way will be the child who acquires healthy self-discipline later.”
—Selma H. Fraiberg (20th century)
“There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)