 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 consecutive colossally abundant numbers is prime, extending Ramanujan's results on the quotients of consecutive superior highly composite number.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.