Harry Altman's webpage
My name is Harry Altman. I've been studying some combinatorial number theory
involving the complexity of computing natural numbers in some simple
computational models, and I've also done some research on computing some things
with ordinals, and, well, various other weird stuff.
I can be reached via email at . (If you can't see my email address, that's because it's been
can find it ROT13ed in there.) My PGP public key can be found
Math papers and drafts
- Integer Complexity, Addition Chains, and
Well-Ordering -- My dissertation.
- A note: This dissertation takes the form of four papers "stapled together",
together with an introduction and a conclusion. Chapter 2 is "Numbers with
Integer Complexity Close to the Lower Bound", above. Chapter 3 is "Integer
Complexity and Well-Ordering", above. Chapter 4 is "Internal Structure of
Addition Chains: Well-Ordering." Chapter 5 has been split into "Integer
Complexity: Representing Numbers of Bounded Defect" and "Integer Complexity:
Algorithms and Computational Results".
- Another note: Some of the problems in Chapter 6 have actually already been
solved, either by myself or by Juan Arias de Reyna... more on that some other
- My implementation of the described algorithms
- Written in Haskell. See the readme for usage instructions. A warning: I
really haven't made the code very readable, or put the functions in a sensible
order. I have at least made a point of describing what each of them do, though.
Polished drafts, should be readable
Nothing to go here at the moment.
Old drafts, may be incomprehensible and out of date
Numbers with integer complexity close to the
lower bound I (with Joshua Zelinsky): In which we devise a method for
classifying numbers whose integer
complexity is close to the lower bound, and derive various
consequences. This is out of date and not very readable.
- Classification of n with
d(n)≤22d(2): The table of results from the computation described in
the above draft. Currently not included in the draft proper, as even in this
compressed form it's 7 pages long. We have since come up with a way of doing
these computations algorithmically, so this is no longer fully necessary. Also
I may not have corrected the few mistakes in it.
- Numbers with integer complexity close to the
lower bound: A rather longer version of the above paper in which we
discuss some things we didn't have room for in the short version. Currently in
the process of being turned into multiple papers which will actually be
readable. Some of the conjectures in here have since been proved.
Highest few sums and products of ones: A write-up of
an idea I don't intend to publish -- the method doesn't generalize; my newer
results with Joshua Zelinsky extend and obsolete it. Still I think the method
is interesting and was worth writing up. If you want a better presentation of
the results themselves, proved via a different method, see "Integer Complexity:
The Integer Defect" above.
- Here is the Haskell code used for the computations.
Sometimes informal bloggy writing on problems I don't necessarily know much of
anything about; sometimes questions I asked on MathOverflow to no satisfactory
Things I wrote for a former job
representation in Solidity -- just how does the Solidity language (as of
version 0.8.20) represent data, under the hood? Written for work but possibly
of more general interest.
Here are miscellaneous files I wish to put in public view:
Here are some drawings that people have given me which I have bothered to scan:
And here are some assorted other images.