Let K be a number field of degree n, OK its ring of integers, and DK the absolute value of the discriminant of K. Then the Euclidean minimum of K is the smallest μ such that for any x in K, there exists y in OK such that the absolute value of the norm of x-y is less than μ; that is, there's always an algebraic integer "within" M(K) of every element of K. If M(K) < 1 then OK is Euclidean (this is essentially a rephrasing of the definitin); if M(K) > 1 then OK isn't Euclidean; if M(K) = 1 we don't know. Then we have
Minkowski's conjecture: M(K) ≤ 2-n (Dk)1/2
for totally real number fields K of degree n. (I'm taking this from Eva Bayer-Fluckiger and Gabriele Nebe, On the Euclidean minimum of some real number fields, Journal de Théorie des Nombres de Bordeaux 17 (2005), 437-454, with some rephrasing.)
Curtis McMullen recently proved this for n=6 (link is to a preprint "Minkowski's conjecture, well-rounded lattices and topological dimension", on his web page). This follows proofs:
- for n=2, Minkowski, circa 1900;
- for n=3, Remak, 1928;
- for n=4, Dyson, 1948;
- for n=5, Skubenko, 1976;
and so it seems like we get one higher degree every thirty years! This leads me to formulate the following:
Meta-Minkowski conjecture: Minkowski's conjecture is true, but will never be proven.
(Sketch of heuristic argument: in approximately the year 1840 + 30n, the degree-n case will be proven. So if we wait long enough, Minkowski's conjecture will be proven for any finite value of n.)
Per a question that was asked at the talk, it seems like the approach in each of these papers was substantially different. McMullen gives sufficient conditions for proving Minkowski's conjecture for any given value of n, and it turns out that those conditions were already known for n = 6.
I wonder to what extent it's possible to predict the mathematical future from the mathematical past, in the case of problems such as this where there's a numerical measure of ``incremental" progress. Often I've seen tables where an improving sequence of constants is given for some problem (for example, a sequence of improving constants in some inequality). It's sort of a Moore's Law for mathematical problems.
When would I make a conjecture that a proof for all n would be found? If, say, we had proofs of some statement for n = 1 in 1960, n = 2 in 1990, n = 3 in 2000, n = 4 in 2005, and in general n = k in 2020 - 60/k; then I'd guess that the full problem gets solved in 2020. Of course, in a case like this the proofs would pile on top of each other, and in reality one would be likely to see large numbers of values of n disappearing in one fell swoop sometime in the 2010s.