Colossally abundant numbers are one of several classes of integers that try to capture the notion of having lots of divisors. For a positive integer n, the sum-of-divisors function σ(n) gives the sum of all those numbers that divide n, including 1 and n itself. Paul Bachmann showed that on average, σ(n) is around π²n / 6. Grönwall's theorem, meanwhile, says that the maximal order of σ(n) is ever so slightly larger, specifically there is an increasing sequence of integers n such that for these integers σ(n) is roughly the same size as eγnlog(log(n)), where γ is the Euler–Mascheroni constant. Hence colossally abundant numbers capture the notion of having lots of divisors by requiring them to maximise, for some ε > 0, the value of the function
over all values of n. Bachmann and Grönwall's results ensure that for every ε > 0 this function has a maximum and that as ε tends to 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.
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 managed to show that the ratio of two consecutive superabundant numbers was always a prime number, but couldn't quite achieve this for colossally abundant numbers. They did conjecture that this was the case and showed that it 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.
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 (sequence A073751 in OEIS). Alaoglu and Erdős's conjecture would also mean that no value of ε gives four different integers n as maxima of the above function.
Read more about this topic: Colossally Abundant Number
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