zero these maxima will increase. Thus there are infinitely many colossally abundant numbers, although they are rather sparse, with only 22 of them less than 1018. Just like with superior highly composite numbers, an effective construction of the set of all colossally abundant numbers is given by the following monotonic mapping from the positive real numbers. Let for any prime number p and positive real . Then is a colossally abundant number. For every ε the above function has a maximum, but it is not obvious, and in fact not true, that for every ε this maximum value is unique. Alaoglu and Erdős studied how many different values of n could give the same maximal value of the above function for a given value of ε. They showed that for most values of ε there would be a single integer n maximising the function. Later, however, Erdős and Jean-Louis Nicolas showed that for a certain set of discrete values of ε there could be two or four different values of n giving the same maximal value. In their 1944 paper, Alaoglu and Erdős conjectured that the ratio of two consecutive colossally abundant numbers was always a prime number. They showed that this would follow from 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 numbers t for which both pt and qt are rational are the positive integers. Using the corresponding result for three primes—a special case of the six exponentials theorem that Siegel claimed to have proven—they managed to show that the quotient of two consecutive colossally abundant numbers is always either a prime or a semiprime (that is, a number with just two prime factors). The quotient can never be the square of a prime. Alaoglu and Erdős's conjecture remains open, although it has been checked up to at least 107. If true it would mean that there was a sequence of non-distinct prime numbers p1, p2, p3,... such that the nth colossally abundant number was of the form Assuming the conjecture holds, this sequence of primes begins 2, 3, 2, 5, 2, 3, 7, 2 . Alaoglu and Erdős's conjecture would also mean that no value of ε gives four different integers n as maxima of the above function. Relation to the Riemann hypothesis In the 1980s Guy Robin showed