26, biprimes and dimensional gates - how interesting is your age?

Whenever my - or a friend's - birthday is approaching, I try to find something mathematically interesting about the age the person is about to become. It takes the edge off growing old and I discover everyone is secretly nerdier than they let on. Easy choices are primes, numbers with lots of factors, squares and so on, and I recently made my way through each of those as I turned 23, 24 and 25. Sometimes there are more exciting options. 3, 7, 31 are Mersenne primes. 4 is 2², 27 is 3³, and you're unlikely to make it to 4⁴ (sorry). 6 is a perfect number, which explains why A.A. Milne wanted to be 6 for ever and ever, but he may have lacked the clever-as-cleverness to realise 7 is happy. Apparently perfection doesn't equate to happiness... until you turn 28, that is.

But what can you say about 26, my latest age?

Well, it has just two prime factors, 2 and 13, which makes it a 'biprime' (or 'semiprime'). In some sense, that's the closest you can get to primality without hitting it. That's kind of interesting, but is it as special as being prime? If you know n primes then you can get n(n+1)/2 biprimes right off the bat by multiplying together every pair of primes and removing the ones you counted twice, so there are more known biprimes. But perhaps that's unfair: we're letting ourselves count biprimes over a much larger range, which might include unknown primes (and biprimes) as well. If, on the other hand, we consider the ratio of primes/biprimes up to and including a certain number k, the primes put up a better fight. In fact, for the first couple of dozen values of k the primes are more common, with a few points where the frequencies match. And the first value of k at which biprimes outnumber the primes? 26! 26 is the hero (or villain) that starts the revolution. Nice one, 26.

We see two more flips at 31 and 34, where the balance of power changes again. I think it would be fun to have an infinite chain of these numerical nemeses. Sadly the k = 41 to 120 rows are all green, suggesting that perhaps the biprimes do eventually overpower the primes forever (at k = 120 there are 31 primes to 38 biprimes). Not that that's a given: processes like random walks are capable of arbitrarily large excursions away from their starting points, yet still return to the beginning infinitely often. At any rate there might still be a limiting ratio that would give a notion of relative scarcity (hence 'specialness'!).

I only discovered the above mini-accolade for 26 as I sat down to write this entry, so I wasn't aware of it previously. Biprimality in itself not seeming too special, I figured I was in for a boring year until I happened across something in the book Fermat's Last Theorem by Simon Singh. There is only one instance of a square number being exactly two fewer than a cube, i.e.

m² = n³ - 2,

for some integers m and n. The square and cube in question are 25 = 5² and 27 = 3³, which means that 26 is uniquely sandwiched exactly above a square and below a cube. Half the fun of recreational mathematics is sexing up the names of everything you encounter, so let's call 26 the 'dimension gate', since it bridges the gap between the second and third dimensions. What about numbers

d = m² = n³

at which the dimensions truly meet, such as 64 = 8² = 4³, 729 = 27² = 9³, 1,000,000 = 1,000² = 100³? I think these have a more natural dimension-crossing quality than 26, so maybe they're more like wormholes. Imagine the topology caused by squares and cubes (and more) stretching and squishing into each other all over the number line; there's something very sci-fi about it all!

Finally, for the algebra-lovers out there (and in here), 26 is the number of sporadic groups.

So the next time you're dreading the prospect of becoming another year older, see if you can work out something exciting and unique about your new age. It won't make you any younger (who are we kidding - it'll probably do the opposite), but at least all those times you're forced to confront that number, your reaction might be 'ooh' rather than 'argh'.

Happy mathsing!