Sodoku Prime – dancing the Latin Square


I’ll be the first to admit – I’m slightly obsessed with trying to come up with a solution to factor ridiculously large semiprimes. I’m not really a mathematician, so I have no hope of solving it, but its great for eating up any spare time I have.

Most of the time, I keep quiet about anything I discover on my journey. Getting from p=xy and p=s² + r to p = x² + 2x sqrt((r + (s-x)²/2x)² + 2s(r + (s-x)²/2x) -r)) might be exciting to me, but it’s hardly the stuff that makes for great dinner party conversations. Quite honestly, I’m sure it’s the sort of thing that real mathematicians crank out twice before breakfast.

Occasionally, however, I’ll stumble across something so simple and beautiful – so pure – that I feel compelled to share it with everyone.

Like the discovery that made me change the way that I felt about Prime numbers, forever.

Before, I’d always thought of Prime numbers as lonely and solitary. Numbers that were indivisible. Numbers for which there was no discernible pattern, no reason or rhyme.

As I played around with different bases, however, I noticed something quite beautiful.

To explain, let’s look at a base that everyone is familiar with – base ten.

John Napier was a mathematician, who amongst various other things five centuries or so ago, invented a device called Napier’s bones to be used in multiplication.

You probably recognise that as a version of the ten times table. What’s important, however are all the numbers under the dividing bar – or as you probably call them – the units.

Note that there’s no real pattern to their overall distribution. The number 7, for example, appears only 4 times, whilst 2 appears 12 times.

Now. Take a look at base 7.

The units for base 7 work just like a sudoku puzzle. You’ll find 1 to 6 under the bar in every line exactly once, horizontal or vertical. The mathematical name for that sudoku-like pattern is a Latin Square, and the beautiful thing that I discovered is that every base which is a prime number works exactly like that.

Prime numbers aren’t as mysterious and aloof as I imagined them to be. There is rhyme and reason to them. There is internal structure and harmony at their core.

It’s the non-prime number bases which are actually chaotic.

And that flipped my entire worldview of prime numbers totally upside down.