Activity 2-14: The ABC Conjecture

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www.carom-maths.co.uk
Activity 2-14 The ABC Conjecture
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 The square-free part of a number is the
largest square-free number that divides into
it. 
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.
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.
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Task can you find rad(n) for n 25 to 30?
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25 52, rad(25)526 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
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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?
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1. Now try A 1, B 8.   2. Now try A 3, B
125.   3. Now try A 1, B 512.
1. gives D 0.666   2. gives D 0.234...
  3. gives D 0.222...
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It has been proved by the mathematician Masser
that D can be arbitrarily small.That means
given any positive number e, we can find numbers
A and B so that D lt e.
See what this means using the
ABC Conjecture spreadsheet
http//www.s253053503.websitehome.co.uk/carom/caro
m-files/carom-2-17.xls
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Smallest Ds found so far
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The ABC conjecture says
has a minimum value greater than zero whenever
n is greater than 1.
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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
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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
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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
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Some consequences of the ABC Conjecture if true
ThueSiegelRoth theorem on diophantine
approximation of algebraic numbers Fermat's Last
Theorem for all sufficiently large exponents
(already proven in general by Andrew Wiles)
(Granville 2002) The Mordell conjecture (already
proven in general by Gerd Faltings) (Elkies
1991) The ErdosWoods conjecture except for a
finite number of counterexamples (Langevin
1993) The existence of infinitely many
non-Wieferich primes (Silverman 1988) The weak
form of Marshall Hall's conjecture on the
separation between squares and cubes of integers
(Nitaj 1996) The FermatCatalan conjecture, a
generalization of Fermat's last theorem
concerning powers that are sums of powers
(Pomerance 2008) The L function L(s,(-d/.))
formed with the Legendre symbol, has no Siegel
zero (this consequence actually requires a
uniform version of the abc conjecture in number
fields, not only the abc conjecture as formulated
above for rational integers) (Granville
2000) P(x) has only finitely many perfect powers
for integral x for P a polynomial with at least
three simple zeros.2 A generalization of
Tijdeman's theorem concerning the number of
solutions of ym xn k (Tijdeman's theorem
answers the case k 1), and Pillai's conjecture
(1931) concerning the number of solutions of Aym
Bxn k. It is equivalent to the
GranvilleLangevin conjecture, that if f is a
square-free binary form of degree n gt 2, then for
every real ßgt2 there is a constant C(f,ß) such
for all coprime integer x,y, the radical of
f(x,y) exceeds C.maxx,yn-ß.34 It is
equivalent to the modified Szpiro conjecture,
which would yield a bound of rad(abc)1.2e
(Oesterlé 1988). And others
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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.Oct 2014 still being
verified...Wikipedia
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With thanks toIvars Peterson's MathTrekand
Wikipedia
http//www.maa.org/mathland/mathtrek_12_8.html
Carom is written by Jonny Griffiths,
hello_at_jonny-griffiths.net