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Day #10: ABC Conjecture

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abc-1

Shinichi Mochizuki

Q: Can you see the equivalence of the three following statements of the ABC conjecture??…

ABC Conjecture I. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that:

c>\operatorname {rad} (abc)^{1+\varepsilon }.

ABC Conjecture II. For every ε > 0, there exists a constant Kε such that for all triples (a, b, c) of coprime positive integers, with a + b = c:

  {\displaystyle c<K_{\varepsilon }\cdot \operatorname {rad} (abc)^{1+\varepsilon }.}

ABC Conjecture III. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε.

Note: q(a, b, c) of the triple (a, b, c), defined as
q(a,b,c)={\frac {\log(c)}{\log(\operatorname {rad} (abc))}}.
For example,
q(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820…
q(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565…
A typical triple (a, b, c) of coprime positive integers with a + b = c will have c < rad(abc), i.e. q(a, b, c) < 1. Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers.

A: Sure you do!

 

The abc Conjecture proved by Shinichi Mochizuki

The abc conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985). It is stated in terms of three positive integers, a, b and c (hence the name) that are relatively prime and satisfy a + b = c. If d denotes the product of the distinct prime factors of abc, the conjecture essentially states that d is usually not much smaller than c. In other words: if a and b are composed from large powers of primes, then c is usually not divisible by large powers of primes. The precise statement is given below. See also What´s new.

Lucien Szpiro attempted a solution in 2007, but it was found to be incorrect.[1] In August 2012 Shinichi Mochizuki posted his four preprints which develop a new inter-universal Teichmüller theory, with an alleged application to the proof of several famous conjectures including the abc conjecture. His papers were submitted to a mathematical journal and are being refereed, while various activities to study his theory have been run. Many mathematicians remain skeptical of his work, and it may take years for the question to resolved due to the strangeness of his proof, and other difficulties like Mochizuki’s earlier resistance to leaving Japan to explain his work to others.[2]

Formulations

Before we state the conjecture we need to introduce the notion of the radical of an integer: for a positive integern, the radical of n, denoted rad(n), is the product of the distinct prime factors of n. For example

rad(16) = rad(24) = 2,
rad(17) = 17,
rad(18) = rad(2 ⋅ 32) = 2 · 3 = 6.

If a, b, and c are coprime[3] positive integers such that a + b = c, it turns out that “usually” c < rad(abc). The abc conjecture deals with the exceptions. Specifically, it states that:

ABC Conjecture I. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that:

c>\operatorname {rad} (abc)^{1+\varepsilon }.

An equivalent formulation states that:

ABC Conjecture II. For every ε > 0, there exists a constant Kε such that for all triples (a, b, c) of coprime positive integers, with a + b = c:

  {\displaystyle c<K_{\varepsilon }\cdot \operatorname {rad} (abc)^{1+\varepsilon }.}

A third equivalent formulation of the conjecture involves the qualityq(a, b, c) of the triple (a, b, c), defined as

q(a,b,c)={\frac {\log(c)}{\log(\operatorname {rad} (abc))}}.

For example,

q(4, 127, 131) = log(131) / log(rad(4·127·131)) = log(131) / log(2·127·131) = 0.46820…
q(3, 125, 128) = log(128) / log(rad(3·125·128)) = log(128) / log(30) = 1.426565…

A typical triple (a, b, c) of coprime positive integers with a + b = c will have c < rad(abc), i.e. q(a, b, c) < 1. Triples with q > 1 such as in the second example are rather special, they consist of numbers divisible by high powers of small prime numbers.

ABC Conjecture III. For every ε > 0, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε.

Whereas it is known that there are infinitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1, the conjecture predicts that only finitely many of those have q > 1.01 or q > 1.001 or even q > 1.0001, etc. In particular, if the conjecture is true then there must exist a triple (a, b, c) which achieves the maximal possible quality q(a, b, c) .

Examples of triples with small radical

The condition that ε > 0 is necessary as there exist infinitely many triples a, b, c with rad(abc) < c. For example let:

{\displaystyle a=1,\quad b=2^{6n}-1,\quad c=2^{6n},\qquad n>1.}

First we note that b is divisible by 9:

  {\displaystyle b=2^{6n}-1=64^{n}-1^{n}=(64-1)(\cdots )=9\cdot 7\cdot (\cdots )}

Using this fact we calculate:

{\displaystyle {\begin{aligned}\operatorname {rad} (abc)&=\operatorname {rad} (a)\operatorname {rad} (b)\operatorname {rad} (c)\\&=\operatorname {rad} (1)\operatorname {rad} \left(2^{6n}-1\right)\operatorname {rad} \left(2^{6n}\right)\\&=2\operatorname {rad} \left(2^{6n}-1\right)\\&=2\operatorname {rad} \left(9\cdot {\tfrac {b}{9}}\right)\\&\leqslant 2\cdot 3\cdot {\tfrac {b}{9}}\\&=2{\tfrac {b}{3}}\\&<{\tfrac {2}{3}}c&&b<c\end{aligned}}}

By replacing the exponent 6n by other exponents forcing b to have larger square factors, the ratio between the radical and c can be made arbitrarily small. Specifically, let p > 2 be a prime and consider:

{\displaystyle a=1,\quad b=2^{p(p-1)n}-1,\quad c=2^{p(p-1)n},\qquad n>1.}

Now we claim that b is divisible by p2:

  {\displaystyle {\begin{aligned}b&=2^{p(p-1)n}-1\\&=\left(2^{p(p-1)}\right)^{n}-1^{n}\\&=\left(2^{p(p-1)}-1\right)(\cdots )\\&=p^{2}\cdot r(\cdots )&&p^{2}|2^{p(p-1)}-1\end{aligned}}}

And now with a similar calculation as above we have:

  {\displaystyle \operatorname {rad} (abc)<{\tfrac {2}{p}}c}

A list of the highest-quality triples (triples with a particularly small radical relative to c) is given below; the highest quality, 1.6299, was found by Eric Reyssat (Lando & Zvonkin 2004, p. 137) for

a = 2,
b = 310·109 = 6,436,341,
c = 235 = 6,436,343,
rad(abc) = 15042.

The abc conjecture has already become well known for the number of interesting consequences it entails. Many famous conjectures and theorems in number theory would follow immediately from the abc conjecture. Goldfeld (1996) described the abc conjecture as “the most important unsolved problem in Diophantine analysis“.

Source: WikipediA

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