A quick thought on addition and multiplication

[sniffnoy]

EDIT: Clarity.

Here's another entry from the file.

It's an old idea that addition and multiplication of whole numbers don't get along very well. I attended a talk by my advisor recently on the subject. But one thing that struck me from the examples he used is that the idea of "addition and multiplication don't get along very well" seems to take two distinctly different flavors.

One of his examples was the abc conjecture, and some related ideas; another was Gödel's incompleteness theorem. But the first of these, essentially, says that addition predictably destroys multiplicative structure. While the second says that the interaction of addition and multiplication is so tangled as to be unpredictable.

(His third example was my own well-ordering result regarding the defects of integer complexity, but honestly I'm not sure it even fits into this category at all. The set of defects has some really nice structure! But that gets into stuff (due to a combination of Juan Arias de Reyna and myself) I haven't really talked about here and probably won't get to for a while.)

Anyway, I don't really know where I'm going with this. I think my point just is, "Addition and multiplication don't get along" seems to really be two different ideas actually.

-Harry

Comments

[joshuazelinsky.livejournal.com]

Wow, That's a really nice compliment to you by Lagarias that he thought that that the well-ordering of defects was a good example of this to go up there with ABC and similar things. It seems that the defects example is more similar to what is happening with ABC. The integer complexity result most connected to this sort of addition and multiplication not going well together would seem more that the integer complexity of n sometimes doesn't arise from the most obvious representation of n (e.g. how there are primes such that |p| !- 1+|p-1|).

[sniffnoy.livejournal.com]

Yeah, I know, right? :)

Actually part of the point of the talk seemed to be "let me promote some stuff I and people I know are doing" -- he talked about ABC, but more of that time was talking about the related XYZ conjecture of himself and Soundararajan.

Did the Gödel stuff lead somewhere else? Hm. Not that I recall.

EDIT: I somehow left the "del" out of "Gödel".