Four Exponentials Conjecture - Corollaries

Corollaries

Using Euler's identity this conjecture implies the transcendence of many numbers involving e and π. For example, taking x₁ = 1, x₂ = √2, y₁ = iπ, and y₂ = iπ√2, the conjecture — if true — implies that one of the following four numbers is transcendental:

The first of these is just −1, and the fourth is 1, so the conjecture implies that eiπ√2 is transcendental (which is already known, by consequence of the Gelfond–Schneider theorem).

An open problem in number theory settled by the conjecture is the question of whether there exists a non-integral real number t such that both 2t and 3t are integers, or indeed such that at and bt are both integers for some pair of integers a and b that are multiplicatively independent over the integers. Values of t such that 2t is an integer are all of the form t = log₂m for some integer m, while for 3t to be an integer, t must be of the form t = log₃n for some integer n. At present it is unknown if there exist integers m and n, not both equal to 1, such that log₂m = log₃n. By setting x₁ = 1, x₂ = t, y₁ = log2, and y₂ = log3, the four exponentials conjecture implies that if t is irrational then one of the following four numbers is transcendental:

So if 2t and 3t are both integers then the conjecture implies that t must be a rational number. Since the only rational numbers t for which 2t is also rational are the integers, this implies that there are no non-integral real numbers t such that both 2t and 3t are integers. It is this consequence, for any two primes not just 2 and 3, that Alaoglu and Erdős desired in their paper as it would imply the conjecture that the quotient of two colossally abundant numbers is prime, extending Ramanujan's results on the quotients of consecutive superior highly composite number.