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 βij are four algebraic numbers for 1 ≤ i,j ≤ 2 such that the following four numbers are algebraic:
then xi yj = βij 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.
Read more about this topic: Four Exponentials Conjecture
Famous quotes containing the words sharp and/or conjecture:
“‘A parted ev’n just between twelve and one, ev’n at the
turning o’ the tide; for after I saw him fumble with the
sheets, and play with flowers, and smile upon his finger’s
end, I knew there was but one way; for his nose was as sharp as
pen, and ‘a babbled of green fields.”
—William Shakespeare (1564–1616)
“What these perplexities of my uncle Toby were,—’tis impossible for you to guess;Mif you could,—I should blush ... as an author; inasmuch as I set no small store by myself upon this very account, that my reader has never yet been able to guess at any thing. And ... if I thought you was able to form the least ... conjecture to yourself, of what was to come in the next page,—I would tear it out of my book.”
—Laurence Sterne (1713–1768)