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. 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. After decades of pretty much stagnation in constructive quantum field theory, it might indicate (like 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” deciding whether an approach is indeed solving the problem. a mass gap in given Euclidean quantum field theories probably does not exist because it is not the right model. 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.
Why is the odd perfect number problem a good problem? First of all, it is very 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. 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 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 (distributed computing) and having connections with cryptology.
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The Landau problems
(a 4*3*3 poem from Oct 23, 2018)
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Goldbach: Every integer 2n>2
is a sum
of two primes.
Twin prime: There are infinitely
many primes p,
p+2, prime twins.
Legendre: Between two consecutive
perfect squares, there
is a prime.
Landau: There are infinitely
many primes of
the form n^2+1.
----------------------------------------In the presentation, I also give some reason, why I think that there are odd perfect numbers. I like guiding principles, even if they are not rigorous (and are obviously not so) like Occam’s razor. They still give you a compass. 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 simply can not imagine a “creator” building a world which is not optimal on a fundamental level. It is the good old “best of all possible worlds” feel which Leibniz formulated in 1710. For me as a mathematician it is just a guiding principle which allows to select an area where to work in. I’m not so much interested in Navier Stokes for example, because it feels like an artificial problem. It is not fundamental because on a basic level, a fluid or gas is not made of continuum stuff, but of discrete particles. The question of global solutions is a rather artificial one. I’m also not interested so much in the two body problem for general relativity simply because it is not the right problem; mass points move on geodesics in the metric and in the same time shape the metric via the Einstein equations. Like electro-hydrodynamics, where charged particles moving around contribute to the magnetic field and so produce a complicated link of ordinary and partial differential equations, the problem of two bodies in general relativity already requires to do some rather ugly numerical tricks like Newtonian approximations. It is not a decoupled system of two different variational principles, it is a coupled system and that variational principle is not formulated yet. It is still a “human made” and not a “divine problem”.. The later would give a simple mechanism allowing to deal with the two body problem in GR. For now, we only are able to approach it with ugly hacks. The odd perfect number problem on the other hand is a divine problem. It is inevitable and is certainly considered by any mathematician living in other parts of the universe, even millions of light years away from us (provided there should be mathematicians elsewhere than on earth). It is rather doubtful, whether mathematicians in an other galaxy would consider Navier Stokes. They might look at some discrete lattice gas approach for example or simply simulate it as a multi-scale n-body problem, similarly as the PIC (particle in cell) method we use here on earth (my wife Ruth had been using the PIC on Cray super computers and I could in gradate school also try out to run some Fortran on that machine, I computed orbits of the standard map ….).
We need some guiding principles anyway, as exhibited in the talk. These are sometimes called “hypothesis” like the Church-Turing thesis. We need for example to assume that we have a reliable memory bank in our head. Assume for example that we do not function properly in basic logic, then we might fool ourselves that we are doing the right thing. Like a person suffering from Alzheimer, we would have a fundamental problem even in assessing that we can not think properly. We also make the assumption that everything we can compute can be computed in the way we know to compute which is usually formulated in the form of the Turing machine. This is the Church-Turing thesis. It would be awful if there were other computations possible. The simplest alternative is something a religious person might try: Just ask your god for the answer. That certainly would be faster and more efficient than a Turing machine! I myself have not had any luck with such experiments. No god has helped me to compute impossibly hard things for example. And there are plenty of problems one can bounce off in a prayer: for example “please factor the 5th Fermat prime for me!”. As I once explained in a yearbook letter to the class of 2020, I have ruled out the existence of supernatural creatures like a devil by experiment. An other mechanism which would fool us is if we were in a simulation, something which we never can rule out (we might be able to figure out that we are in a simulation but then that game designer who we are able to locate could again be just part of a simulation and so on). So, we might just be fooled to believe that we can think logically and the programmer could change that without us noticing that. If you for example would lose the ability to have a memory that works, we can not even form conclusions any more. We derive A, then work on something else and derive B. Now, we should be able to conclude (A and B), but only if we did not forget about A. Well, we have on earth many examples of entities which work as such. Most politicians say A, then B, and then Not A and then Not B and nobody notices!
Now to the guiding principle which let me think that there are odd perfect numbers, or why the Riemann hypothesis is probably false, or why the Collatz problem is probably false (I believe that it is likely that there be periodic orbits of the 3x+1 map for very, very large numbers). All these predictions are based on the principle that “if there is no mechanism which prevents something, it will happen”. It is the feel that the validity of some phenomenon must come with a reason. This can be some symmetry or an invariant or some cancellation or some insight. It can also be a guiding principle to at least (“with the stomach”) get a feel for whether a proof is valid or not. In the case of the last theorem of Fermat for example, the proof came with considerable insight and connections with other mathematical fields like elliptic curves. The claimed proof of the ABC conjecture however does not come with sufficient additional insight, which makes the result look fishy to me (as a non-expert). Most wrong proofs (and there are many) prove something by building a large mess or prove things indirectly and building a large computation mess which somewhere a tiny mistake is hidden, leading to the desired contradiction.
[Added March 18, 2025: the no-impediment principle is of course a mathematical principle. In real life we have other principles. If there is no reason for somebody to be not guilty, it does not mean that the person is guilty. The mathematical principle reflects that mathematics huge. If there is no law or symmetry or invariant that prevents something to happen, then it is a solid assumption that it probably won’t happen or reversed if there is no impediment for something to happen, it probably happens. A good example is really the Collatz 3x+1 problem. There are many reasons (probabilistical for example) which suggest that the chance of a number to escape to infinity under the Collatz 3x+1 rule is zero. In average the dynamics points down. Also, the chance of running into a periodic loop far far away is slim and gets slimmer and slimmer as larger the numbers become. But the chance is not zero (at least we do not know of a number theoretical impediment for it to happen). It is like waiting for the big payoff in the petersburg paradox pushing the game to a win. In the Petersburg casino, we know that it does not happen in our life time. The Poincare recurrence does not happen for any practical reasons but it happens in mathematics. We have a mathematical theorem that tells us that we end up with a half empty chamber in a divided chamber. If we assume ergodicity, then it will happen eventually that have of the room in our office will be empty of particles. It is only for practical reasons (we would have to wait for googol’s of ages of the universe for this to happen) that we dismiss it. Mathematically, we can not dismiss it because it a theorem. Simple games like the Collatz 3x+1 problem or the perfect number problem can illustrate well the discrepancy between “physics” and “mathematics”. In physics, we accept the second law of thermodynamics, even so it is a mathematical absurdity colliding with basic mathematics like Poincare recurrence and reversibility. All truly fundamental laws in physics we know are perfectly reversible. There are more realistic models in thermodynamics which assume entropy to increase but these models have transitioned from micro canonical ensembles to canonical ensembles, where we disregard the individual motions of particles. Mathematics is much larger than physics. The monkey typing randomly illustrates that everything we ever write or prove or think is contained in any single random sequence. Mathematical reality is so much larger than physical reality. The “no impediment principle” = “if there is no reason for something to be true, it is probably false” or “if there is no reason for something to be false, it is probably true” is a reasonable rule to go by, at least when being in the dark about what to make about a mathematical issue. It can be applied to any open problem. The Goldbach conjecture is probably true because there are no symmetries in the primes which make them look different than a random set of numbers with that probabilistic distribution. The prime twin conjecture is probably true again because we have no reason to see why there should be a minimal gap of 4 or larger between two primes. The Legendre conjecture also: once we have passed the small number impediments that there might be a large gap of primes beating the gap between squares, we are more and more likely to get a prime between two squares: the distribution is on our side. Finally also the Landau problem whether there are infinitely many primes of the form n^2+1, why should there not be infinitely many. We might have a hard time to prove this (and currently no mathematical artillery can attack that problem) but there would have to be a truely miraculous new rule about primes implying that only finitely many primes are of the form n^2+1. ]
Back to the “odd perfect number” problem, there is no mechanism known which prevents odd perfect numbers to exist. There are lots and lots of mechanisms present however which make the occurrence very, very unlikely meaning that we have to “wait a long time” to find one. This has the effect that a counter example might be very very large. There are many examples in mathematics, where such phenomena exist. Look at the Petersburg paradox for example. You can have a life as long as an elf in the lord of the rings and still not win in the Petersburg paradox, even so the expectation value is known to be infinite (you win 2^n with probability 2^(-n)) Mathematics shows that the motion of particles in a fluid are perfectly deterministic and reversible. Take a two chamber box however with a gas in one part and a vacuum in the other and build a connection. The gas will get to an equilibrium with equal pressure on both sides. Mathematically however, we know that there will be a time, when all particles are again in the single chamber and no particle in the other. Mathematical certainty and physical certainty are completely different things. Physics even has a second law of thermodynamics which “proves” that this “recurrence” can not occur. For any physicist or practical person, the recurrence is impossible. For a mathematician however it is not. Indeed, all proofs people have cooked up to prove the second law are flawed. They are proofs showing that from a practical point of view it does not work, but we mathematicians can imagine a Maxwell demon made of completely different elementary particles which can beat the second law. One can argue that it does not exist because there are no such particles but that is not an argument. And since in principle we could simulate a world in which some “demons” exist made of particles of size 
