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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?
2
5,
25 =
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 subdisciplines 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
With thanks to:
Ivars Peterson's MathTrek
http://www.maa.org/mathland/mathtrek_12_8.html
Carom is written by Jonny Griffiths, [email protected]