www.carom-maths.co.uk Activity 2-14: The ABC Conjecture

A square-free number is one that is not divisible by any square except for 1. So 35713 = 1365 is square-free. So 335472132 = 139741875 is not square-free.  The ‘square-free part’ of a number is the largest square-free number that divides into it.   This is also called ‘the radical’ of an integer n. To find rad(n), write down the factorisation of n into primes, and then cross out all the powers.

Task: can you find rad(n) for n = 25 to 30?

25 = 52, rad(25)=5 26 = 213, rad(26)=26 27 = 33, rad(27)=3 28 = 227, rad(28)=14 29 = 29, rad(29)=29 30 = 235, rad(30)=30

Task: now pick two whole numbers, A and B, whose highest common factor is 1. (This is usually written as gcd (A, B) = 1.)   Now say A + B = C, and find C. Now find D = Do this several times, for various A and B. What values of D do you get?

1. Now try A = 1, B = 8.   2. Now try A = 3, B = 125. 3. Now try A = 1, B = 512. 1. gives D = 2   2. gives D = 0.234... 3. gives D = 0.222...

It has been proved by the mathematician Masser that D can be arbitrarily small. That means given any positive number ε, we can find numbers A and B so that D < ε. Try to see what this means using the ABC Excel spreadsheet

The ABC conjecture says; has a minimum value greater than zero whenever n is greater than 1.

‘Astonishingly, a proof of the ABC conjecture would provide a way of establishing Fermat's Last Theorem in less than a page of mathematical reasoning. Indeed, many famous conjectures and theorems in number theory would follow immediately from the ABC conjecture, sometimes in just a few lines.’ Ivars Peterson

‘The ABC conjecture is amazingly simple compared to the deep questions in number theory. This strange conjecture turns out to be equivalent to all the main problems. It's at the centre of everything that's been going on. Nowadays, if you're working on a problem in number theory, you often think about whether the problem follows from the ABC conjecture.’ Andrew J. Granville

‘The ABC conjecture is the most important unsolved problem in number theory. Seeing so many Diophantine problems unexpectedly encapsulated into a single equation drives home the feeling that all the sub-disciplines of mathematics are aspects of a single underlying unity, and that at its heart lie pure language and simple expressibility.’ Dorian Goldfeld

Stop Press!!! In August 2012, Shinichi Mochizuki released a paper with a serious claim to a proof of the abc conjecture. Mochizuki calls the theory on which this proof is based inter-universal Teichmüller theory, and it has other applications including a proof of Szpiro's conjecture and Vojta's conjecture. Wikipedia

Carom is written by Jonny Griffiths, mail@jonny-griffiths.net With thanks to: Ivars Peterson's MathTrek http://www.maa.org/mathland/mathtrek_12_8.html Carom is written by Jonny Griffiths, mail@jonny-griffiths.net