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New math post on my blog: What does it mean to expand a function “in powers of x-1”?

New math post on my blog: What does it mean to expand a function “in powers of x-1”?

I really hate this tired claim: “This is confusing because \(0.\bar 01\) seems to indicate a decimal with "infinite zeros and then a one at the end." Which, of course, is absurd.” Sure, it's not a real number, but there's nothing intrinsically wrong with an infinite sequence of zeroes followed by a one.

Bill Gasarch asks: why is there no grid for Hilbert's 10th? blog.computationalcomplexity.o

What he wants to know is, for which pairs (d,n) can we algorithmically find integer solutions to degree-d n-variable polynomial equations, and for which pairs is it undecidable. The answer seems to be: we can solve them when d ≤ 2, we can't solve them for some pairs of larger numbers, and there's a big gap of unknown pairs.

If you want to go the jargon route, you can distill it down to once sentence, something like “Take an irreducible polynomial X of degree k over the finite field GF(p) and compute the quotient ring GF(p)[x] /‹X›, what's so hard about that?”
But of course such an answer is useless to almost anyone who doesn't already know it.

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Someone on Math SE asked how to calculate the multiplication table of a finite field, and I wrote up a brief but sufficient explanation at a fairly low jargon level. Might be good for undergrads or advanced high school students.

Sir Thomas Urquhart was a 17th-century Scottish eccentric who tried to systematize a new language for trigonometry; the law of sines was abbreviated as “eproso”, which (if you know the system) encapsulates its meaning.

Easy but fun problem I found in an ancient Math SE post: What is the smallest integer greater than 1 such that ½ of it is a perfect square and ⅕ of it is a perfect fifth power?

Found in Friedman's _A History of Folding in Mathematics_, p. 71, a quote from Francesco Maurolico from 1537: "Item manifestum est in unoquoque regularium solidorum, numerum basium coniunctum cum numero cacuminum conflare numerum, qui binario excedit numerum laterum".

Except for the fact that he considers only Platonic solids, this is Euler's formula V-E+F=2 for convex polyhedra (in the equivalent form V+F=E+2), long before Euler (1752) and Descartes (1630).

Housebound advice from a long-term housebound person, boosts ok 

I'm going to post a long thread on how to cope better if you're stuck indoors at the moment. I've been housebound by disability for a long while now, and sporadically housebound due to depression and agoraphobia before that. These are things that have improved how I deal with that. This is not gonna make houseboundness painless for you but it might help you cope. Disabled comrades are welcome to share their own advice to this thread.
> *normal mathematician*: Look at this theorem I proved!
> *category theorist*: By thinking deeply about your result I was able to prove a far-reaching generalization of it.
> *normal mathematician*: Cool! What are some other examples of it?
> *category theorist*: …

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