Fri, 21 Jan 2022
Divisibility and modular residues are among the most important concepts in elementary number theory, but the terminology for them is clumsy and hard to pronounce.
The first two are 8 syllables long. The last one is tolerably short but is backwards. Similarly:
is awful. It can be abbreviated to
but that is also long, and introduces a dummy !!k!! that may be completely superfluous. You can say “!!n!! is !!3!! mod !!5!!” or “!!n!! mod !!5!! is !!3!!” but people find that confusing if there is a lot of it piled up.
Common terms should be short and clean. I wish there were a mathematical jargon term for “has the form !!5k+3!!” that was not so cumbersome. And I would like a term for “mod-5 residue” that is comparable in length and simplicity to “fifth root”.
For mod-!!2!! residues we have the special term “parity”. I wonder if something like “!!5!!-ity” could catch on? This doesn't seem too barbaric to me. It's quite similar to the terminology we already use for !!n!!-gons. What is the name for a polygon with !!33!! sides? Is it a triskadekawhatever? No, it's just a !!33!!-gon, simple.
Then one might say things like:
For “multiple of !!n!!” I suggest that “even” and “odd” be extended so that "!!5!!-even" means a multiple of !!5!!, and "!!5!!-odd" means a nonmultiple of !!5!!. I think “!!n!! is 5-odd” is a clear improvement on “!!n!! is a nonmultiple of 5”:
It's conceivable that “5-ity” could be mistaken for “five-eighty” but I don't think it will be a big problem in practice. The stress is different, the vowel is different, and also, numbers like !!380!! and !!580!! just do not come up that often.
The next mouth-full-of-marbles term I'd want to take on would be “is relatively prime to”. I'd want it to be short, punchy, and symmetric-sounding. I wonder if it would be enough to abbreviate “least common multiple” and “greatest common divsor” to “join” and “meet” respectively? Then “!!m!! and !!n!! are relatively prime” becomes “!!m!! meet !!n!! is !!1!!” and we get short phrasings like “If !!m!! is !!n!!-even, then !!m!! join !!n!! is just !!m!!”. We might abbreviate a little further: “!!m!! meet !!n!! is 1” becomes just “!!m!! meets !!n!!”.
[ Addendum: Eirikr Åsheim reminds me that “!!m!! and !!n!! are coprime” is already standard and is shorter than “!!m!! is relatively prime to !!n!!”. True, I had forgotten. ]