Perfect Number Bomb - Quantum Calculus
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Quantum Calculus<br>Calculus without limits
This spring, scientific american was asking around in departments about problems mathematicians are thinking about. I immediately wrote back mentioning the "odd perfect number problem ", a problem which has for strange reasons been snubbed by problem collections. The article in SCIAM has now appeared. As I had once taken the topic of odd perfect numbers as a theme in a mathcircle talk [PDF] (see also the slides [PDF] and media [M4V]), I will not get into the mathematics and culture of the problem but commented in a movie about the reason, why I believe that there are odd perfect numbers and that we will find them within the next 100 years. This was also an opportunity to vent a bit about the choice of problem collections and about general principles on how to feel about whether something is reasonable or not and finally on how one would go about finding odd perfect numbers.
Problem lists are always subjective of course. Hilbert’s list of 23 problems contained some great problems but also some terrible choices. As for the most famous open problem list, the Millenium problems , most choices are rather obvious. My own grief is with the Yang-Mills problem which is terribly imprecise for my taste. I have never worked in that field but the mass gap problem just does not feel right. The reason is that it does not specify what mathematical model can be allowed and what would it mean to be a solution. After decades of pretty much stagnation in constructive quantum field theory, it might indicate (like Freeman Dyson once hinted) that this is just a rather artificial frame work and that the model can not explain properly fundamental phenomena like confinement or the masses of fundamental particles. My real concerns about this problem is that it is a "political problem", where one would need an "authority" (meaning a decorated VIP mathematician) deciding whether an approach is indeed solving the problem. A mass gap in the given Euclidean quantum field theories probably does not exist because it is most likely not the right model (some of the best mathematicians have worked on it and not succeeded. In mathematical physics, there are much better problem collections, like the 15 problems of Simon , which are all much better than the ambiguous Yang-Mills problem. Also Hilbert is guilty of terrible problems like asking for an axiom system for physics.
Why is the odd perfect number problem a "good problem"? First of all, it is very, every clear. Like the Landau problems, the question can be understood in middle school : add up all the proper factors of a number. If the sum is the number itself, it is called perfect. the problem does not even need the definition of primes. The only mathematical sophistication is the concept of "factor" which we get familiar with when learning the multiplication table in first grade. It is hard to understand why the problem is undervalued. The concept of odd perfect numbers have come up as one of the first things in mathematics. We do not know exactly when the question has really first been formally been asked but it belongs certainly to the earliest mathematical problems which have been considered . The even perfect numbers are also very interesting as they are related to the largest known primes . Searching for large primes is a large collaborative project (GIMPS ) and interesting not only for mathematicians. It is also a computational and so computer science phenomenon (a large scale experiment in distributed computing) and has connections with cryptology, as modern security depends on number theoretical difficulties like factoring integers or finding discrete logs in finite Abelian groups. I once summarized for a mathtable talk on Goldbach comets the four problems of Landau as a poem:
The Landau problems<br>(a 4*3*3 poem from Oct 23, 2018)<br>Goldbach : Every integer 2n>2<br>is a sum<br>of two primes.<br>Twin prime : There are infinitely<br>many primes p,<br>p+2, prime twins.<br>Legendre : Between two consecutive<br>perfect squares, there<br>is a prime.<br>Landau : There are infinitely<br>many primes of<br>the form n^2+1.
In the presentation, I also give some reason, why I think that there are odd perfect numbers . I also comment on meta thinking like guiding principles, even if they are not rigorous (and are obviously and almost by definition not so) like Occam’s razor (one of my favorites also in political issues. [Politics is for mathematicians very frustrating as different folks operate with different definitions and different value functions and different signals (information resources) and form complicated multi-player situations.] They still give you a compas s. Besides ‘simplicity" one of the most important guiding principles is "beauty" . Many mathematicians are guided by that and it usually pays off. If something is too complicated, it is probably just the wrong approach. Maybe it feels a bit like a religion but I...