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\title[Two ``case studies'' on Islamic mathematics ]{Two ``case studies'' on Islamic mathematics}

\author
{John B. Little}

\institute[College of the Holy Cross]
{
  MATH 110-2 -- Algebra Through History\\
}

\date[November, 2019]
{November, 2019}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}
{\bf Outline}

\tableofcontents

\end{frame}

\section{Goals for Today}

\begin{frame}
{Getting here from there}

\begin{enumerate}
\item Try to get some evidence regarding the claims of historians like Morris
Kline that Islamic mathematics only ``transmitted'' Greek knowledge to 
Europe without adding anything very important or original
\item Look at a particular family of questions in number theory
\item Trace history from Euclid through the contributions of several Islamic
mathematicians; see what we know now
\item (More briefly) to consider some work of Umar al-Khayyami (Omar Khayyam)
on solving cubic equations
\end{enumerate}
\end{frame}

\section{Background}

\begin{frame}
{\bf Back with the Pythagoreans}

\begin{enumerate}
\item In early Greek mathematics the Pythagorean brotherhood
\item very much ``into'' number symbolism
\item the ``tetraktys,'' triangular numbers, squares, pentagonal numbers, ... , 
``perfect numbers,'' ``amicable pairs of numbers,'' etc.
\item part of their philosophy of the way numbers and mathematical relationships
were part of the order underlying the cosmos
\end{enumerate}
\end{frame}

\begin{frame}
{\bf A bit of number theory from the {\it Elements}}

\begin{enumerate}
\item In Proposition 36 of Book IX of the {\it Elements}, Euclid studied \emph{perfect numbers}
\item[] 
\begin{defn}  A number $n$ is said to be \emph{perfect} if it equals the sum of all of its proper divisors 
(``aliquot parts'' -- the whole numbers less than $n$ that divide $n$ evenly)
\end{defn}
\item For example, $n = 6$ is a perfect number (the smallest one) since the proper divisors of $6$ are
$1,2,3$ and $1 + 2 + 3 = 6$
\item Similarly $n = 28$ is perfect since its proper divisors are $1,2,4,7,14$ and 
$1 + 2 + 4 + 7 + 14 = 28$
\end{enumerate}
\end{frame}

\begin{frame}
{\bf Euclid's result}

\begin{thm}[{\it Elements} IX, 36]  
A number $n = 2^{p-1}\times (2^p - 1)$ is perfect if and only if 
$2^p - 1$ is prime.
\end{thm}

\begin{enumerate}
\item Recall, primes are numbers $p$ whose only divisors are $1$ and the number itself: $p = 2,3,5,7,11,13,17,19, ... $
\item Euclid had shown in Book IX, 20 that there are infinitely many primes (one of the jewels of this section of 
the {\it Elements})
\item Not too hard to see that $2^p - 1$ can only be prime if $p$ itself is prime
\end{enumerate}
\end{frame}

\begin{frame}
{\bf Perfect numbers}

\begin{enumerate}
\item For example, $2^2 - 1 = 3$, $2^3 - 1 = 7$, $2^5 - 1 = 31$, $2^7 - 1 = 127$ are prime
\item But $2^{11} - 1 = 2047 = 23 \cdot 89$ is not prime  
\item So by Euclid IX, 36, for instance $n = 2^{3 - 1}\times (2^3 - 1) = 28$, $n = 2^{5-1}\times (2^5 - 1) = 496$, 
$n = 2^{7-1}\times (2^7 - 1) = 8128$
are perfect numbers  
\item But $2^{11-1} \times (2^{11} - 1) = 2096128$ is not perfect
\end{enumerate}
\end{frame}

\section{Work from the Islamic world}

\begin{frame}
{\bf Ibn al-Haytham and Thabit ibn-Qurra}

\begin{enumerate}
\item Another Islamic mathematician named Ibn al-Haytham (ca. 1000 CE) (known as ``Alhazen'' in medieval and 
later Europe through a mangling of his real name!) studied this and conjectured that the {\it only time} $n$ is an even perfect
number is when $n$ has Euclid's form $n = 2^{p-1} \times (2^p - 1)$
\item But he was not able to prove this completely
\item Thabit ibn-Qurra (836 - 901 CE), whom we met briefly before, 
was interested in perfect numbers too, and other related questions
\item[]
\begin{defn} Two numbers $m,n$ are said to be an \emph{amicable pair} if 
the sum of the proper divisors of $m$ is $n$ and, vice versa, the sum of the proper
divisors of $n$ is $m$.  
\end{defn}
\end{enumerate}
\end{frame}

\begin{frame}
{\bf The smallest example}

\begin{enumerate}
\item $m = 220$ and $n = 284$ are the smallest amicable pair:
\item The proper divisors of $m = 220 = 2^2 \times 5 \times 11$ are $1,2,4,5,10,20,11,22,44,55,110$ and 
$$1 + 2 + 4 + 5 + 10 + 20 + 11 + 22 + 44 + 55 + 110 = 284,$$ 
while
\item the proper divisors of $n = 284 = 2^2 \times 71$ are $1,2,4,71,142$ and 
$$1 + 2 + 4 + 71 + 142 = 220.$$
\end{enumerate}
\end{frame}

\begin{frame}
{\bf Thabit's theorem}

Thabit was interested in determining whether there was a systematic
way to generate amicable pairs -- something along the lines of Euclid's formula 
for perfect numbers.

\begin{thm}  Suppose that for some integer $k \ge 1$,  $p = 3 \times 2^{k-1} - 1$, $q = 3\times 2^k - 1$
and $r = 9 \times 2^{2k-1} - 1$ are all primes.  Then $m = 2^k \times p \times q$ and $n = 2^k \times r$
are an amicable pair.  
\end{thm}

Example:  $k = 2$ gives $p = 5, q = 11, r = 71$ all prime.  So $m = 4\times 5 \times 11 = 220$ and $n = 4 \times 71 = 284$
are amicable(!)  
\end{frame} 

\begin{frame}
{\it A more balanced view(?)}

\begin{enumerate}
\item The Islamic tradition widened the range of mathematics by enlarging the role 
of numerical and algebraic computation and introducing the number system we still 
use -- not just geometry
\item But these mathematicians also studied the foundations of Euclidean geometry and 
attempted to resolve some of the outstanding questions 
\item Extended work of Archimedes and other Greeks in several ways
\end{enumerate}
\end{frame}

\section{Later developments}

\begin{frame}
{\bf Leonhard Euler}

\begin{enumerate}
\item Alhazen's conjecture about Euclid's formula for perfect numbers
was eventually proved true by Leonhard Euler (1707 - 1783, CE) a Swiss mathematician, one of the 
most prolific and original mathematicians of the modern era, built on work of Pierre de Fermat,
Mersenne that \emph{rediscovered} the results of Thabit ibn-Qurra, and others.  
\item Might they actually have had access to Thabit's work??  
\item Euler also generalized Thabit ibn-Qurra's method for generating amicable pairs and showed
it could be used to get lots of them!
\end{enumerate}

\end{frame}

\begin{frame}
{\it Mathematics is not ``finished!''}

\begin{enumerate}
\item Students encountering mathematics in school may be tempted to think that 
the subject is ``finished'' and that everything that can be known about it is 
already known
\item But this is {\it far from true}!!!
\item The very questions we are talking about here include some very famous
{\it unsolved problems} in mathematics -- cases where no one knows the answer 
and where, if one of you were to be able to develop an answer, your name
would join the ranks of famous mathematicians.
\end{enumerate}

\end{frame}

\begin{frame}
{\bf For instance, ... }

\begin{enumerate}
\item In Euclid's formula, $n = 2^{p-1}\times (2^p - 1)$,
natural to ask:  Are there infinitely many primes $p$ for which $2^p - 1$ is also a prime 
number?  Or does this generate only finitely many different perfect numbers?
\item Answer:  No one knows!  Such primes are called ``Mersenne primes'' after a later
French monk and mathematician Marin Mersenne (1588 - 1648, CE) who studied them
\item At the current time (as of November, 2019), exactly 
51 Mersenne primes are known to exist.  The largest 
is $2^{82,589,933}-1$
\item Found via extensive computer calculations including massive computations on 
distributed networks of computers over the internet -- GIMPS project: 
can donate your computer's free time to the search if you want!
\end{enumerate} 
\end{frame}

\begin{frame}
{\bf For instance, ... }

\begin{enumerate}
\item In Euclid's formula for perfect numbers, $n = 2^{p-1}\times (2^p - 1)$, is always
{\it even}, so when $n$ is perfect, we only get even perfect numbers this way.
\item Question:  Are there any \emph{odd} perfect numbers?
\item Answer:  Nobody knows!!
\item Some indications that the answer is probably ``no,'' but nothing definitive
yet.  Any progress here would basically make a mathematician's career!
\end{enumerate} 
\end{frame}

\begin{frame}
{\bf For instance, ... }

\begin{enumerate}
\item Similarly, one can ask:  Are there infinitely many amicable pairs?
\item Answer:  Nobody knows!!
\item By using Thabit's formulas, Euler's generalizations, lots of computing, 
many such pairs are known (in the millions!) but again, nothing definitive.
\end{enumerate} 
\end{frame}

\section{al-Khayyami and cubic equations}
\begin{frame}
{\bf Umar al-Khayyami and cubic equations}

\begin{enumerate}
\item Umar made a systematic study of classes of cubic equations 
\item Very similar in spirit to what Al-Khwarizmi had done for 
quadratic equations
\item E.g. "A cube plus sides equals numbers" was one case
\item In modern notation, $x^3 + cx = d$
\item Still assuming $c, d > 0$ here, so for instance, al-Khayyami
would treat $x^3 = cx + d$ as a different type of equation.
\end{enumerate}
\end{frame}

\begin{frame}
{\bf al-Khayyami's solution}

\begin{enumerate}
\item Heavily based on properties of conic sections -- work of Apollonius
\item Apollonius' {\it Conics} had been translated from Greek to Arabic some time before and
was a standard reference for mathematicians at this time.  
\item The solution of the cubic equation $x^3 + cx = d$ was phrased in {\it geometric terms}.
\item In modern language:
\end{enumerate}
\end{frame}

\begin{frame}
{\bf The solution (in modern terms), continued}

\begin{enumerate}
\item Take the number $\frac{d}{2c}$ and construct a circle with 
that radius and center at $\left(\frac{d}{2c},0\right)$.
\item The equation of that circle is $x^2 - \frac{d}{c}x + y^2 = 0$
\item Also construct the parabola with equation $y = \frac{1}{\sqrt{c}} x^2$.
\item Then substituting for $y$ in the equation of the circle, 
$$x^2 - \frac{d}{c}x + \frac{1}{c} x^4 = 0$$
so rearranging and canceling one power of $x$,
$$x^3 + cx = d\qquad (Pretty\ amazing!)$$
\end{enumerate}
\end{frame}

\begin{frame}
{\it Conclusions}
\begin{enumerate}
\item The other cases are treated using some of the same ideas.  
\item This was definitely an advance over what the Greeks had done before,
\item and over
what others were doing at this time in Europe.  
\end{enumerate}
\end{frame}

\end{document}

