Breaking a number down to primes, and why the result is unique

0:00Suppose you want to know what two numbers have in common. The honest way is to list every factor of each, and read the two lists against one another. For thirty and forty-two that is eight factors and eight factors, and it does work. But you had to find all sixteen to use four. And the lists do not grow in any way you can predict ahead of time. Below a hundred, the most factors any number has is twelve, and five different numbers reach it.

0:32Ninety-six is one of those five. Ninety-seven, sitting right beside it, has two. So you cannot tell how long the job is until you have done it. We need a better technique. Start from the one thing that stops the process. Most numbers break into two smaller factors. Thirty is six times five. Ninety is nine times ten. Some will not. Seven has nothing but one and itself, so there is no way to write it as two factors above one.

1:06Those are the primes, and the useful way to hold them here is simply: the numbers that will not split. Everything else is composite, which only means it still splits. So here is a procedure. Take any number, split it, split the pieces, and keep going. Every step makes the pieces smaller, so it cannot run forever. And when it stops, everything left standing is a prime. Take ninety. There are five ways to break it into two factors, and any one of them will do.

1:42Take three times thirty. Three will not split. Thirty splits into two times fifteen. Two will not split. Fifteen splits into three times five. And now nothing on the board can go any further. Reading them off in the order they arrived: three, two, three, five. Multiply those back together and you get ninety. That is a prime factorisation. Now do it again from a different split, and watch carefully.

2:15Begin with two times forty-five this time. Two is finished. Forty-five is nine times five, and nine is three times three. That leaves two, three, three, five. Or begin with nine times ten, which shares no number at all with either of the first two routes. Nine gives three and three. Ten gives two and five. Three, three, two, five. Three different journeys. The order came out different every time. The primes did not: one two, two threes, one five, every time.

2:52That is worth stopping on, because everything after this depends on it. If the primes you finish with depend on how you started, a factorisation is just one description of a number among many. If they do not, it is the description, and things can be read straight off it. So let us be exact about what has actually been established. Ninety has five starting splits, and every route from every one of them ends at the same four primes.

3:25I ran the same check on every number up to two hundred and fifty-nine, and it never once failed. Two hundred and forty can be taken apart sixty-nine different ways. All sixty-nine finish in the same place. That is a great deal of evidence. It is still not a proof, and the difference is worth ten honest seconds. Here is why that difference matters, and it is the best thing in this topic.

3:55The sameness is not something the splitting gives you for free. It is a fact about whole numbers. To see that, shrink the world. Keep only the numbers that leave remainder one when you divide by four. One, five, nine, thirteen, seventeen, twenty-one, twenty-five, and onwards. Inside that world nine will not split, because three is not there to split it with. Nor will twenty-one, or forty-nine. They are the unsplittable numbers of this smaller world.

4:28Now take four hundred and forty-one. It is nine times forty-nine. It is also twenty-one times twenty-one. Same procedure, run to a standstill, two genuinely different answers. So when our numbers finish the same way every time, that is a real fact about them, not a triviality. One tidy-up before we go on. What is the prime factorisation of seven? It is seven. That looks like a dodge. It is not.

5:02The procedure says split until nothing splits, and at seven you are already finished, so the answer is a list with one thing on it. Every prime behaves that way. Thirteen is thirteen. Ninety-seven is ninety-seven. It matters because it lets the general statement carry no exceptions at all. Every number above one is a product of primes, and for a prime itself that product simply has one factor in it.

5:35There is a neat way to write the process down, and it is worth learning to read. Take one hundred and five, and put it in a circle. To its left write a prime that divides it. Three. Underneath, write what is left after that division. Thirty-five. Circle that as well. Now the same move again. Five to the left of thirty-five, and seven underneath. Seven will not split, so it stays bare, and the diagram is finished.

6:06The rule holding the whole thing together is this: every circled number equals the one to its left times the one below it. One hundred and five is three times thirty-five. Thirty-five is five times seven. Check that, and the diagram has checked itself. And now the move that turns a diagram into an answer. Sweep down the left-hand column, and round the bottom. Three, then five, and then seven standing at the foot.

6:40One hundred and five is three times five times seven. The left column and the last entry are the whole answer. Nothing else in the picture was ever needed. Which means the circles can come off. Divisor on the left, the number beside it, the quotient on the line below, and the last quotient alone at the bottom. That is not a second method. It is the same method with the drawing rubbed out.

7:11So let us spend it on something bigger. Twelve hundred. One route starts with forty times thirty, because those two you can see at a glance. Forty is eight times five, and eight is two times two times two. Thirty is six times five, and six is two times three. It gets there. But you are tracking a spreading tree of pieces while you do it. The other route just keeps dividing. Two into twelve hundred is six hundred. Two into six hundred is three hundred.

7:45Two into three hundred is a hundred and fifty. Two into that is seventy-five. Three into seventy-five is twenty-five. Five into twenty-five is five. Six divisions, and the answer is the column you wrote going down: four twos, a three, and two fives. Both are correct. One of them you can do without thinking, and on a large number that is the whole difference. Here is a picture to finish on, and it spends everything we just built.

8:21A hundred rings, one for each number from one to a hundred. Every ring is cut into arcs, and every arc is coloured. There is no key. Work it out. Thirty-five is one orange arc and one blue. Forty-nine is two blues. Fifty is one green and two oranges. So blue is seven, orange is five, green is two, and the number of arcs is the number of primes, counting repeats.

8:50Thirty-six is two greens and two magentas, which makes magenta three. But nothing is coloured for eleven, or thirteen, or anything larger. They all share red, which is how red ends up on fifty-four of the hundred rings, more than any other colour. And the most divided rings are not the largest numbers. Sixty-four and ninety-six are cut into six arcs each, while ninety-seven is one unbroken circle. Twenty-six of the hundred rings are unbroken: the twenty-five primes, and the number one, which has no primes inside it at all.

9:34Every one of the other seventy-four is built out of those, and out of nothing else. Which raises the obvious question. How big does a prime get? The largest one anybody has found turned up in October of twenty twenty-four. Written out in full it runs to forty-one million, twenty-four thousand, three hundred and twenty digits. At one digit a second, without ever stopping, writing it out would take about a year and a third.

10:07And that number will not split. Nothing divides it but one and itself. Every number you will ever meet is built out of primes, put together in exactly one way. Which is why that word exactly was worth the care.