Algebraic Independence of Known Constants
Although both and e are known to be transcendental, It is not known whether the set of both of them is algebraically independent over . In fact, it is not even known if is irrational. Nesterenko proved in 1996 that:
- the numbers π, eπ, and Γ(1/4) are algebraically independent over Q.
- the numbers π, eπ√3, and Γ(1/3) are algebraically independent over Q.
- for all positive integers n, the numbers π, eπ√n are algebraically independent over Q.
Read more about this topic: Algebraic Independence
Famous quotes containing the words algebraic and/or independence:
“I have no scheme about it,no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (18171862)
“We must have constantly present in our minds the difference between independence and liberty. Liberty is a right of doing whatever the laws permit, and if a citizen could do what they forbid he would no longer be possessed of liberty.”
—Charles Louis de Secondat Montesquieu (16891755)