Activity 2-14: The ABC Conjecture

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 3 3  5 4  7 2  13 2 = 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.

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

25 = 5 2, rad(25)=5 26 = 2  13, rad(26)=26 27 = 3 3, rad(27)=3 28 = 2 2  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 = Now try A = 3, B = Now try A = 1, B = gives D = 0.666… 2. gives D = gives D =

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. ABC Conjecture spreadsheet See what this means using the carom/carom-files/carom-2-17.xls

Smallest Ds found so far…

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

Some consequences of the ABC Conjecture if true… Thue–Siegel–Roth theoremThue–Siegel–Roth theorem on diophantine approximation of algebraic numbers Fermat's Last TheoremFermat's Last Theorem for all sufficiently large exponents (already proven in general by Andrew Wiles) (Granville 2002)Andrew WilesGranville 2002 The Mordell conjecture (already proven in general by Gerd Faltings) (Elkies 1991)Mordell conjectureGerd FaltingsElkies 1991 The Erdős–Woods conjecture except for a finite number of counterexamples (Langevin 1993)Erdős–Woods conjectureLangevin 1993 The existence of infinitely many non-Wieferich primes (Silverman 1988)non-Wieferich primesSilverman 1988 The weak form of Marshall Hall's conjecture on the separation between squares and cubes of integers (Nitaj 1996)Marshall Hall's conjectureNitaj 1996 The Fermat–Catalan conjecture, a generalization of Fermat's last theorem concerning powers that are sums of powers (Pomerance 2008)Fermat–Catalan conjecturePomerance 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)L functionLegendre symbolSiegel zeroGranville 2000 P(x) has only finitely many perfect powers for integral x for P a polynomial with at least three simple zeros. [2]integralpolynomial [2] A generalization of Tijdeman's theorem concerning the number of solutions of y m = x n + k (Tijdeman's theorem answers the case k = 1), and Pillai's conjecture (1931) concerning the number of solutions of Ay m = Bx n + k.Tijdeman's theorem It is equivalent to the Granville–Langevin conjecture, that if f is a square-free binary form of degree n > 2, then for every real β>2 there is a constant C(f,β) such for all coprime integer x,y, the radical of f(x,y) exceeds C.max{|x|,|y|} n-β. [3][4] [3][4] It is equivalent to the modified Szpiro conjecture, which would yield a bound of rad(abc) 1.2+ε (Oesterlé 1988).Szpiro conjectureOesterlé 1988 And others…

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

With thanks to: Ivars Peterson's MathTrek and Wikipedia Ivars Peterson's MathTrek Carom is written by Jonny Griffiths, mathland/mathtrek_ 12_8.html