How do we not already have words for this stuff? The adjectives "even" and "odd" describe numbers of the form 2k and 2k+1 with integer k, respectively. What are the equivalents for moduli other than 2?
I've looked around, and I can't even find any proposals out there. It might not seem like a big deal, but I often see math teachers stumble trying to explain concepts related to modular arithmetic, when every noun has to have a k and a plus sign in it. In this video, for example, Dr. Burkard Polster of Mathologer had to resort to defining and using the adjectives "good" and "bad" for primes of the form 4k+1 and 4k+3, respectively.
And in the everyday world, we use the words "even" and "odd" all the time to describe periodic patterns and occurrences, but when the period in question is anything except two, we usually just don't bother. If we have to, we'll use clunky phrases like "numbers that are one less than a multiple of four", but you'll certainly raise some eyebrows among non-math folk if you're casually throwing around "k"s in your speech. Neologisms have been created for far less than this.
So here's my proposal. For whole numbers m ≥ 2 and r < m, the adjective meaning "of the form mk+r with integer k" will be comprised of the following two elements, concatenated:
- The remainder prefix: none when r = 0, "un" when r = 1, and when r ≥ 2, take the word for 1000r+1 and remove "llion". Additionally, the prefix "non-" can be used for r ≠ 0.
- The modulus suffix: "even" when m = 2, and when m ≥ 3, take the word for 1000m+1, remove "llion", and add "te".
A couple quick examples. To determine the word for "4k+2", take r = 2 and m = 4.The remainder prefix is taken from the word for 10002+1 = 10003 = a billion; removing the "llion" thus yields "bi". The modulus suffix, likewise, is taken from the word for 10004+1 = 10005 = a quadrillion; removing the "llion" and adding "te" yields "quadrite". So the full word is "biquadrite". For "4k" and "4k+1", the words are "quadrite" and "unquadrite" respectively.
These words will function the same way as "even" and "odd": principally as adjectives, but adding an "s" can turn them into plural nouns. It's not a hard system to learn; in practice, most of the roots are ones we already know instinctively from the words "triple", "quadruple", etc. Here are some sample sentences - for comparison, try to rephrase each sentence in traditional layman's terms and see how much more cumbersome it is:
- All unquadrite primes are sums of two squares, but no triquadrites are.
- A quintiseptite number has a septite number of partitions.
- All primes above three are either unsextite or quintisextite.
- We issue a quarterly report in the first week of every untrite month.
- The olympic games occur in even years; summer in the quadrites, winter in the biquadrites.
- A musical passage in 12/8 without notes on bitrite beats has a swing rhythm.
- The first Soviet economic plans were issued on triquintite years.
- Every quadrite year has a leap day, except for non-quadricentite centites.
The most mutable part of this proposal is the "ite" suffix at the end of these words. I chose this suffix quite arbitrarily, because the words it created were short while still yielding no existing results on OneLook's comprehensive online dictionary search. I also tried the suffix "iven", based on the word "even", and there might be a better one out there with a more solid etymological basis.
The most observant of you might notice a potential ambiguity exposed in the last of those sample sentences. The word "quadricentite", without the "non-" prefix, could mean either "400k" or "100k+4"! There are other such ambiguities with power-of-10 moduli, and I experimented initially with using a mix of Greek and Latin roots to get around this problem, but decided to opt for simplicity and ease of use over airtight clarity. In the case of the leap year sentence, not only does the "non-" prefix resolve the ambiguity, but the context makes it clear as well - if "quadricentite" in this case meant "100k+4", then there would be no such thing as "quadricentite centites", so the entire use of the term would be pointless. In practice, big power-of-10 moduli like these are pretty rare, and when they do come up, you can always just forego the neologisms if necessary.
Another unfortunate drawback to this system is that numbers divisible by 3, including my absolute favourite number, are now "trite". I consider it a personal sacrifice that I'm willing to make for the common good.