Unique factorisation — the series

1. 

   One way to factor, and no other

   Every number breaks into primes in exactly one way. That is so familiar it is hard to see as a claim at all — until it is put beside an arithmetic where it is false, and where six has two different factorisations that cannot be reconciled.

   part 1 · number
2. 

   The shape of a number's divisors

   Lay a number's divisors out as a lattice with one axis per prime, and two of the most useful facts in arithmetic stop being formulas and become the width and the corner of a rectangle.

   part 2 · number
3. 

   A factorisation that hides its primes

   Keep only the whole numbers one more than a multiple of four. They multiply among themselves and nothing is lost — yet 441 is 9 × 49 and also 21 × 21, and every one of those factors is unbreakable there. Unique factorisation turns out not to be a fact about multiplication at all.

   part 3 · number
4. 

   Factoring uniquely with no way to divide

   Unique factorisation is proved by dividing with a small remainder, and in the Gaussian integers that works because discs of radius one cover the plane. In the integers of ℚ(√−19) the discs leave holes, no division algorithm of any kind can be made to work — and factorisation is unique anyway. The same field is why n² + n + 41 is prime forty times running.

   part 4 · number
5. 

   Counting the classes that break factorisation

   The class number measures how badly unique factorisation fails in a field, and defined through ideals it looks impossible to compute. Gauss computed it by hand, for every field he wanted, by counting quadratic forms — and every form can be squeezed, by changes of variable that keep its values, into exactly one small standard shape.

   part 5 · number
6. 

   Euler's sixty-five convenient numbers

   For some n, whether a prime can be written as x² + ny² is settled by its remainder on division by 4n alone, the way a prime's remainder on division by four settles whether it is a sum of two squares. Euler found sixty-five such n, from 1 to 1,848, called them convenient, and used the largest to prove that 18,518,809 is prime. Every one of them is a number whose class group has no element of order more than two — and whether the list is complete is still not known.

   part 6 · number
7. 

   Class groups drawn at random

   The class number of an imaginary quadratic field measures how badly unique factorisation fails there, and it looks arbitrary: 1, 2, 4, 3, 5, 12, 1. Count how often 3 divides it and the answer is not one time in three but noticeably more — heading, by a conjecture of Cohen and Lenstra, for 44 per cent. The reason is that class groups behave like finite groups chosen at random with each group weighted by one over its number of symmetries, so groups with few symmetries, like the cyclic group of order 3, turn up more often than the naive count suggests.

   part 7 · number
8. 

   How many primes a typical number has

   A typical number near N has about log log N different prime factors, and the count is spread around that in a bell curve whose variance is log log N as well. Both facts are theorems. Neither is visible at any size anyone can count: up to ten million the average is right and the spread is less than half what the limit says.

   part 8 · number
9. 

   The same bell on thinner sets

   A typical whole number has about log log N prime factors, spread in a bell curve. That was a theorem about all numbers, built on the picture of each prime dividing independently. The numbers one less than a prime, and the numbers n² + 1, are far too thin for that picture to have any obvious claim on them, and on both the same bell appears — with means shifted by constants that come straight from how often each small prime divides them. The squares plus one that are prime, the case of a single factor, number 102,205 up to n = two million against 102,302 predicted; whether there are infinitely many is open.

   part 9 · number
10. 

    The largest prime in a typical number

    Count a number's prime factors and the answer is about log log n. Ask instead how big the largest one is, and the answer is a power of n with a law of its own: half of all large numbers have a prime factor above n^0.6065, and the share whose primes all stay below n^(1/u) is Dickman's ρ(u), a function defined by its own past. The same law governs the longest cycle of a random shuffle, and there it arrives at once; for the numbers themselves, at ten million, it is still visibly on its way.

    part 10 · number
11. 

    Two neighbours share out their small primes

    A number and the next one have no prime in common, and the natural guess is that their largest primes are independent. In the limit that is believed and nearly proved. Below ten million it is visibly false: neighbours are both made of small primes only nine-tenths as often as independence predicts, a run of four only two-thirds as often, and the correlation between their largest primes is negative at distance one and positive at distance six — because a small prime that divides one of two neighbours is a small prime the other cannot have.

    part 11 · number
12. 

    Most numbers are missing from the table

    An N × N multiplication table has N² cells, and every product in it is less than or equal to N². Yet only 22.5% of the numbers that small turn up in the 10,000 × 10,000 table, and the share keeps falling. Erdős proved in 1955 that it falls to nought; Kevin Ford found in 2008 how slowly — like a tiny power of the logarithm, 0.0861 — because a product of two numbers has too many prime factors to be typical.

    part 12 · number
13. 

    Sums with too few primes

    Add 2 to 6,436,341 and the sum is 23⁵: three numbers with only four primes between them, and their product 15,042 is far smaller than the sum. The abc conjecture says this can happen only so far. Of 606 such triples up to 200,000, only seven reach quality 1.4, and if the conjecture holds, Fermat's last theorem and much else follows for large exponents in a line.

    part 13 · number

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