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\begin{document}

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\section*{abc conjecture}

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{masons_theorem}{Mason's Theorem}\dotfill \pageref*{masons_theorem} \linebreak
\noindent\hyperlink{relation_to_other_statements}{Relation to other statements}\dotfill \pageref*{relation_to_other_statements} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{abc conjecture} (or \emph{ABC conjecture}) is a [[number theory|number theoretic]] conjecture due to (\hyperlink{Oesterl&#233;Masser}{Oesterl\'e{}-Masser 1985}), which says that there are only [[finite set|finitely many]] integer solutions to the [[equation]]

\begin{displaymath}
a + b = c   \;\;\; for \;\;\;  a,b,c \geq 1 \in \mathbb{N}
\end{displaymath}
(or instead $a+b+c= 0$) if one requires the [[integer numbers]] $a,b,c$ to have no common factor as well as having ``joint power'' greater than a given bound.

Here the \emph{power} of $a,b,c$ is

\begin{displaymath}
P(a,b,c) \coloneqq \frac{log|c|}{log(rad(a\cdot b\cdot c))}
  \,,
\end{displaymath}
where the [[radical]] $rad(n)$ of an [[integer]] $n$ is the product of all its distinct [[prime factors]].

The precise form of the conjecture is:

\begin{utheorem}
\textbf{(abc conjecture)}

For any number $\epsilon\gt 0$ there are only finitely many positive relatively prime ([[coprime integers|coprime]]) [[integer]] solutions $(a,b,c)$ to the [[equation]] $a + b = c$ with power $P(a,b,c)\geq 1+\epsilon$.

\end{utheorem}
According to (\hyperlink{Matzur}{Mazur}):

\begin{quote}%
The beauty of such a Conjecture is that it captures the intuitive sense that triples of numbers which satisfy a linear relation, and which are divisible by high perfect powers, are rare; the precision of the Conjecture goads one to investigate this rarity quantitatively. Its very statement makes an attractive appeal to perform a range of numerical experiments that would test the empirical waters. On a theoretical level, it is enlightening to understand its relationship to the constellation of standard arithmetic theorems, conjectures, questions, etc., and we shall give some indications of this below.


\end{quote}
\hypertarget{masons_theorem}{}\subsection*{{Mason's Theorem}}\label{masons_theorem}

According to \hyperlink{Lang}{Lang}, one important antecedent of the abc conjecture is a simple but at the time unexpected relation for the function field case, published in 1984. Consider polynomials $f \in k[t]$ over an algebraically closed field $k$ of characteristic $0$, and define $n_0(f)$ to be the number of \emph{distinct} roots of $f$, counted \emph{without} regard to multiplicity.

\begin{utheorem}
Let $a, b, c \in k[t]$ be relatively prime polynomials, not all constant, such that $a + b = c$. Then $\max \{deg(a), deg(b), deg(c)\} \leq n_0(a b c) - 1$.

\end{utheorem}
\begin{proof}
Let $f = a/c$, $g = b/c$, so that $f + g = 1$. Taking the derivative, we obtain

\begin{displaymath}
\frac{f'}{f} f + \frac{g'}{g} g = 0
\end{displaymath}
whence

\begin{displaymath}
b/a = g/f = -\frac{f'/f}{g'/g}.
\end{displaymath}
Put

\begin{displaymath}
a(t) = c_1 \prod (t - \alpha_i)^{m_i}, \qquad b(t) = c_2 \prod (t - \beta_j)^{n_j}, \qquad c(t) = c_3 \prod (t - \gamma_k)^{p_k}.
\end{displaymath}
Then

\begin{displaymath}
\frac{b}{a} = -\frac{f'/f}{g'/g} = -\frac{\sum \frac{m_i}{t - \alpha_i} - \sum \frac{p_k}{t - \gamma_k}}{\sum \frac{n_j}{t - \beta_j} - \sum \frac{p_k}{t - \gamma_k}} .
\end{displaymath}
A common denominator for $f'/f$ and $g'/g$ is given by

\begin{displaymath}
N_0 = \prod (t - \alpha_i) \prod (t - \beta_j) \prod (t - \gamma_k)
\end{displaymath}
whose degree is $n_0(a b c)$. We then have

\begin{displaymath}
\frac{b}{a} = -\frac{N_0 f'/f}{N_0 g'/g}
\end{displaymath}
where the numerator and denominator on the right are polynomials. However, since $b$ and $a$ are relatively prime, the fraction $b/a$ is already in lowest terms. From this we conclude that $deg(b) \leq deg(N_0 f'/f) \leq n_0(a b c) - 1$, and similarly $deg(a) \leq deg(N_0 g'/g) \leq n_0(a b c) - 1$, which completes the proof.

\end{proof}
\begin{ucor}
Assume $x, y, z \in k[t]$ are relatively prime polynomials, not all constant, and suppose $x^n + y^n = z^n$. Then $n \leq 2$.

\end{ucor}
\begin{proof}
From Mason's theorem, we conclude $n deg(x) = deg(x^n) \leq deg(x) + deg(y) + deg(z) - 1$, and similarly upon replacing $x$ by $y$ and $z$ on the left. Adding the results, we have

\begin{displaymath}
n(deg(x) + deg(y) + deg(z)) \leq 3(deg(x) + deg(y) + deg(z)) - 3
\end{displaymath}
which is impossible if $n \geq 3$.

\end{proof}
Guided by analogies between the ring of integers and the ring of polynomials in one variable, and building on insights of Mason, Frey, Szpiro, and others, Masser and Oesterl\'e{} were led to formulate the abc conjecture for integers as follows. Again define $N_0(m)$ for $m$ a non-zero integer to be the number of distinct primes dividing $m$.

\begin{itemize}%
\item Conjecture: For all $\epsilon \lt 0$ there exists $C(\epsilon) \lt 0$ such that for relatively prime integers $a, b, c$ satisfying $a + b = c$, we have\begin{displaymath}
\max({|a|}, {|b|}, {|c|}) \leq C(\epsilon)N_0(a b c)^{1+\epsilon}.
\end{displaymath}


\end{itemize}
Of course, this differs from the polynomial case because of the presence of $1+ \epsilon$ in the exponent, but this is a necessary evil. For example, for any $C \gt 0$, we can find relatively prime $a$, $b$, $c$ with $a + b = c$ and ${|a|} \gt C N_0(a b c)$: take $a = 3^{2^n}$, $b = -1$, and observe by repeated application of $x^2 - y^2 = (x-y)(x+y)$ that $c = a + b$ is of the form $2^n d$ for some integer $d$. Taking $n$ sufficiently large, we can easily derive the claimed inequality.

\hypertarget{relation_to_other_statements}{}\subsection*{{Relation to other statements}}\label{relation_to_other_statements}

The abc conjecture implies the [[Mordell conjecture]] (\hyperlink{Elkies}{Elkies}).

It is equivalent to the general form of [[Szpiro's conjecture]].

\begin{itemize}%
\item [[Vojta's conjecture]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The abc conjecture was stated in

\begin{itemize}%
\item Joseph Oesterl\'e{}, David Masser (1985)

\end{itemize}
[[Shinichi Mochizuki]] anounced the proof which the mathematical community perceives as a serious but unchecked claim. See the references at \emph{[[inter-universal Teichmüller theory]]}.

Comments on the proof are at

\begin{itemize}%
\item \emph{[[Mochizuki's proof of abc]]}.


\item MathOverflow, \emph{\href{http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658}{Philosophy behind Mochizuki's work on the ABC conjecture}}



\end{itemize}
An popular account of the problem of the math community checking the proof is in

\begin{itemize}%
\item Caroline Chen, \emph{The Paradox of the Proof} (\href{http://projectwordsworth.com/the-paradox-of-the-proof/}{web})

\end{itemize}
Mason's theorem was presented in

\begin{itemize}%
\item R.C. Mason, Equations over function fields. In \emph{Number Theory, Proceedings of the Noordwijkerhout}, Springer Lecture Notes 1068 (1984), 149-157.

\end{itemize}
Material on Mason's theorem and its relation to the abc conjecture was taken from

\begin{itemize}%
\item Serge Lang, Algebra ($3^{rd}$ Edition), Addison-Wesley (1993), 194-196.

\end{itemize}
See also

\begin{itemize}%
\item [[Barry Mazur]], \emph{Questions about number}, \href{http://www.math.harvard.edu/~mazur/papers/scanQuest.pdf}{pdf} scan

\end{itemize}
\begin{itemize}%
\item PolyMath, \emph{\href{http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture}{ABC conjecture}}


\item Abderrahmane Nitaj, \emph{\href{http://www.math.unicaen.fr/~nitaj/abc.html}{The ABC Conjecture Homepage}}


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Abc_conjecture}{abc conjecture}}



\end{itemize}
The relation to the [[Mordell conjecture]] is discussed in

\begin{itemize}%
\item [[Noam Elkies]], \emph{ABC conjecture implies Mordell}, Int. Math. Research Notices 7 (1991) 99-109

\end{itemize}
The relation to [[Szpiro's conjecture]] is discussed in

\begin{itemize}%
\item Matt Baker (notes taken by William Stein), \emph{Elliptic curves, the ABC conjecture, and points of small canonical height} (\href{http://modular.math.washington.edu/mcs/archive/Fall2001/notes/12-10-01/12-10-01.pdf}{pdf})

\end{itemize}


\end{document}