@tpfto Once upon a time we weren't expected to be the authors AND the typesetters as well. And yes, it's very time-consuming.
@ColinTheMathmo What I don't know, or can't bring to mind, is whether "-1 is a quadratic residue mod p" and "p is a sum of two squares" are related in a direct and important way or if it's really just a coincidince that "p is congruent to 1 mod 4" characterizes both. The usual proofs I know for those two things don't seem to have much to do with each other but maybe I'll learn something here.
In the first case, the more relevant fact is nthat 29 factors in ℤ[i], the Gaussian integers. (If that's what you meant, then apologies for overexplaining; I misunderstood your notation.) Viz., (5+2i)(5-2i). This is a ring with interesting factorizations because not everything is a unit. I can't think of any ring that uses the "square root of -1" in ℤ/41ℤ in the same way.
@ColinTheMathmo Both ℂ and ℤ/41ℤ are fields, so saying that things "factor nontrivially" in either of those systems... is sort of true, depending on what you mean by nontrivial, but not really the point. Everything factors lots of ways in a field, because everything's a unit. For every nonzero x, there is a y such that 29=x*y (mod 41), and that's nothing special about 29. It isn't really taking advantage of the fact that 9² = -1 (mod 41).
@ColinTheMathmo "Imprisoned" gives better scansion for my taste, maintaining consistent dactyls.
FLEA and a
FLY in a
FLUE were im-
WHAT could they
DO? said the...
@nebusj Hurrah! I'm looking forward to a new series.
A nice page of recent writings about abstract strategy games (mostly connection games:
(Clicking the tags at the bottom of the page leads to many more good, unindexed articles)
@11011110 Heh. Not only that, the opening sentence, "A standard definition of what constitutes a ‘snack’ [is] lacking" sounds remarkably like something a mathematician would open a paper with. Probably a paper that would end up in Christian's "Interesting Esoterica" collection.
@11011110 Thanks! I've assigned this short article as reading in my discrete math class for some time; I didn't realize there was a book!
@ColinTheMathmo Heh, yes, just riffing off your own comment.
@ColinTheMathmo Everywhere I look these days, I see signs of division (and they are called obeli)
For some reason, I feel that when the whole and only point of the question is "think about this reasoning," students pause and take some time to do that. If the question asks for a number, then it's like they have free license to skip the time-consuming "reasoning" step, multiply some things, get a number, and move briskly along to multiplying some different things in the next problem.
@ColinTheMathmo My "author bio" for some article or other read "[The author] grew up reading Martin Gardner and continues to grow by reading Martin Gardner."
And it's still spot-on.
Herzlich Willkommen in Minnesota!
The social network of the future: No ads, no corporate surveillance, ethical design, and decentralization! Own your data with Mastodon!