Transcendental Number Theory

The purpose of the paper is to describe and explain a special case of the four exponentials conjecture in transcendental number theory: specifically that for any two distinct prime numbers p and q, the only real number t for which both p to the t and q to the t are rational are the positive integers. This implies another conjecture, the Alaoglu and Erdos’s conjecture regarding colossally abundant numbers, which claims that the ratio of two consecutive colossally abundant numbers is always a prime number. Alaoglu and Erdos showed in their paper that their conjecture followed from the above conjecture, and they were able to use the corresponding result for three primes – a special case of the six exponentials theorem – to show that the quotient of two consecutive colossally abundant numbers is always either a prime or a semiprime.