Goals for Today

Background

Work from the Islamic world Later developments

A “case study” on Islamic mathematics

John B. Little

MATH 110-2 – Algebra Through History

November, 2019

Goals for Today

Background

Work from the Islamic world Later developments

Outline

1 Goals for Today

2 Background

3 Work from the Islamic world

4 Later developments

Goals for Today Background

Work from the Islamic world Later developments

Getting here from there

1 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

2 Look at a particular family of questions in number theory 3 Trace history from Euclid through the contributions of

several Islamic mathematicians 4 See what we know as of know

Goals for Today Background

Work from the Islamic world Later developments

Back with the Pythagoreans

1 In early Greek mathematics the Pythagorean brotherhood 2 very much “into” number symbolism 3 the “tetraktys,” triangular numbers, squares, pentagonal

numbers, ... , “perfect numbers,” “amicable pairs of numbers,” etc.

4 part of their philosophy of the way numbers and mathematical relationships were part of the order underlying the cosmos

Goals for Today Background

Work from the Islamic world Later developments

A bit of number theory from the Elements

1 In Proposition 36 of Book IX of the Elements, Euclid studied perfect numbers

Definition A number n is said to be perfect if it equals the sum of all of its proper divisors (“aliquot parts” – the whole numbers less than n that divide n evenly)

2 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 3 Similarly n = 28 is perfect since its proper divisors are

1, 2, 4, 7, 14 and 1 + 2 + 4 + 7 + 14 = 28

Goals for Today Background

Work from the Islamic world Later developments

Euclid’s result

Theorem (Elements IX, 36)

A number n = 2^{p−1} × (2^{p} − 1) is perfect if and only if 2^{p} − 1 is prime.

1 Recall, primes are numbers p whose only divisors are 1

and the number itself: p = 2, 3, 5, 7, 11, 13, 17, 19, ... 2 Euclid had shown in Book IX, 20 that there are infinitely

many primes (one of the jewels of this section of the Elements)

3 Not too hard to see that 2^{p} − 1 can only be prime if p itself is prime

Goals for Today Background

Work from the Islamic world Later developments

Perfect numbers

1 For example, 2^{2} − 1 = 3, 2^{3} − 1 = 7, 2^{5} − 1 = 31,

2^{7} − 1 = 127 are prime 2 But 2^{11} − 1 = 2047 = 23 · 89 is not prime 3 So by Euclid IX, 36, for instance n = 2^{3−1} × (2^{3} − 1) = 28,

n = 2^{5−1} × (2^{5} − 1) = 496, n = 2^{7−1} × (2^{7} − 1) = 8128 are perfect numbers

4 But 2^{11−1} × (2^{11} − 1) = 2096128 is not perfect

Goals for Today Background

Work from the Islamic world Later developments

Ibn al-Haytham and Thabit ibn-Qurra

1 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 only time n is an even perfect number is when n has Euclid’s form n = 2^{p−1} × (2^{p} − 1)

2 But he was not able to prove this completely 3 Thabit ibn-Qurra (836 - 901 CE), whom we met briefly

before, was interested in perfect numbers too, and other related questions

Definition Two numbers m, n are said to be an 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.

Goals for Today Background

Work from the Islamic world

Later developments

The smallest example

1 m = 220 and n = 284 are the smallest amicable pair: 2 The proper divisors of m = 220 = 2^{2} × 5 × 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 3 the proper divisors of n = 284 = 2^{2} × 71 are 1, 2, 4, 71, 142

and

1 + 2 + 4 + 71 + 142 = 220.

Goals for Today Background

Work from the Islamic world Later developments

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.

Theorem

Suppose that for some integer k ≥ 1, p = 3 × 2^{k−1} − 1, q = 3 × 2^{k} − 1 and r = 9 × 2^{2k−1} − 1 are all primes. Then m = 2^{k} × p × q and n = 2^{k} × r are an amicable pair.

Example: k = 2 gives p = 5, q = 11, r = 71 all prime. So m = 4 × 5 × 11 = 220 and n = 4 × 71 = 284 are amicable(!)

Goals for Today Background

Work from the Islamic world Later developments

A more balanced view(?)

1 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

2 But these mathematicians also studied the foundations of Euclidean geometry and attempted to resolve some of the outstanding questions

3 Extended work of Archimedes and other Greeks in several ways

Goals for Today Background

Work from the Islamic world Later developments

Leonhard Euler

1 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 rediscovered the results of Thabit ibn-Qurra, and others.

2 Might they actually have had access to Thabit’s work?? 3 Euler also generalized Thabit ibn-Qurra’s method for

generating amicable pairs and showed it could be used to get lots of them!

Goals for Today Background

Work from the Islamic world Later developments

Mathematics is not “finished!”

1 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

2 But this is far from true!!! 3 The very questions we are talking about here include some

very famous 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.

Goals for Today Background

Work from the Islamic world Later developments

For instance, ...

1 In Euclid’s formula, n = 2^{p−1} × (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?

2 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

3 At the current time (as of November, 2019), exactly 51 Mersenne primes are known to exist. The largest is 2^{82,589,933} − 1

4 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!

John B. Little  A “case study” on Islamic mathematics

Goals for Today Background

Work from the Islamic world Later developments

For instance, ...

1 In Euclid’s formula for perfect numbers, n = 2^{p−1} × (2^{p} − 1), is always even, so when n is perfect, we only get even perfect numbers this way.

2 Question: Are there any odd perfect numbers? 3 Answer: Nobody knows!! 4 Some indications that the answer is probably “no,” but

nothing definitive yet. Any progress here would basically make a mathematician’s career!

Goals for Today Background

Work from the Islamic world Later developments

For instance, ...

1 Similarly, one can ask: Are there infinitely many amicable

pairs? 2 Answer: Nobody knows!! 3 By using Thabit’s formulas, Euler’s generalizations, lots of

computing, many such pairs are known (in the millions!) but again, nothing definitive.