I got to thinking about unit fractions. This is in the service of a different problem I might get around to talking about. Unit fractions are fractions, yeah. They allow the denominator to be any counting number. The numerator is always 1, that is, the unit. So, like, , , , these are all unit fractions.
Particularly I was thinking about sums of unit fractions. And whether the sum of a particular group of unit fractions is less than or greater than one. Like, is greater than one, sure. But is less than one. But is ? Is there an easy way to tell? I mean easier than addition, which is admittedly pretty simple to start with. Might be fun to spot a straightforward way to do this.
Where my joy in this fun little problem disappears is realizing, oh, of course, this is a number theory problem. Number theory is about studying how numbers work and what they do. It’s full of great questions you can understand even if you don’t know much mathematics. Like, is there a largest prime number? Is every even number larger than two equal to a sum of two prime numbers? “Can you tell at a glance whether a set of unit fractions adds up to more than 1” fits in that line. (A mathematician might clean that up by saying “can you tell by inspection”, but still.)
The trouble with number theory problems is they pretty much break one of two ways. One is that we have an answer and can prove it. The proof is this 12-line thread of argument so tight you can cut yourself on the reducto-ad-absurdum. Allow a couple symbols and you could fit the thing in two tweets. The other way a number theory problem can break is “well, after 120 years of study, we have a 60-page proof that seems to answer a specialized case of this problem, if we assume that the Riemann hypothesis is true”. Or possibly “if we assume the continuum hypothesis is [ true or false ]”. Anyway, there are people who have some doubts about the section between pages 38 and 44.
I haven’t poked around the literature, not even Wikipedia, yet. So I don’t know which kind this is more likely to be. My suspicion is there’s probably some neat 12-line proofs. Unit fractions are terms in the “harmonic series”. This is the number you get by adding together , carrying on with the denominator going through every whole number ever. This series turns out to “diverge”. You go ahead and pick any number you like. I can then pick a finite set of terms from the harmonic series. And my set of terms will add up to something bigger than your number.
And yet other weird stuff happens. Like, pick any string of digits you like. I’ll say ’35’ because I’m writing this sentence at 35 minutes past the hour. Keep the whole harmonic series except for any terms which have the sequence ’35’ in them. So, no , no , no , no . Although is still in. Add up the infinitely many terms that remain. That will “converge’, adding up to some finite number.
So you see I’m looking at a problem that’s in well-explored waters. This makes me also suspect there isn’t a better answer than “just add your fractions up”. If there were, it would probably be a mildly well-known trick used for arithmetic magic tricks. Or, possibly, as an odd trick used to squeeze some other proof down to 12 lines.