Statement
If x1, x2 and y1, y2 are two pairs of complex numbers, with each pair being linearly independent over the rational numbers, then at least one of the following four numbers is transcendental:
An alternative way of stating the conjecture in terms of logarithms is the following. For 1 ≤ i,j ≤ 2 let λij be complex numbers such that exp(λij) are all algebraic. Suppose λ11 and λ12 are linearly independent over the rational numbers, and λ11 and λ21 are also linearly independent over the rational numbers, then
An equivalent formulation in terms of linear algebra is the following. Let M be the 2×2 matrix
where exp(λij) is algebraic for 1 ≤ i,j ≤ 2. Suppose the two rows of M are linearly independent over the rational numbers, and the two columns of M are linearly independent over the rational numbers. Then the rank of M is 2.
While a 2×2 matrix having linearly independent rows and columns usually means it has rank 2, in this case we require linear independence over a smaller field so the rank isn't forced to be 2. For example, the matrix
has rows and columns that are linearly independent over the rational numbers, since π is irrational. But the rank of the matrix is 1. So in this case the conjecture would imply that at least one of e, eπ, and eπ ² is transcendental (which in this case is already known since e is transcendental).
Read more about this topic: Four Exponentials Conjecture
Famous quotes containing the word statement:
“Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.”
—Charles Sanders Peirce (1839–1914)
“The force of truth that a statement imparts, then, its prominence among the hordes of recorded observations that I may optionally apply to my own life, depends, in addition to the sense that it is argumentatively defensible, on the sense that someone like me, and someone I like, whose voice is audible and who is at least notionally in the same room with me, does or can possibly hold it to be compellingly true.”
—Nicholson Baker (b. 1957)
“Children should know there are limits to family finances or they will confuse “we can’t afford that” with “they don’t want me to have it.” The first statement is a realistic and objective assessment of a situation, while the other carries an emotional message.”
—Jean Ross Peterson (20th century)