Abstract
Number theory has a rich ancient tradition, much of it being found in Books VII–IX of Euclid’s Elements. Among the beautiful results in these three books, one finds a proof that there are infinitely many prime numbers, and that if 2n − 1 is a prime then 2n−1(2n − 1) is a perfect number, i.e., it is equal to the sum of its proper divisors.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The first two perfect numbers are 6 and 28.
- 2.
At that time in the Greek world, “arithmetic” (arithmētikē) referred to what we call “the theory of numbers.” Our “arithmetic” the Greeks called “logistic” (logistikē).
- 3.
However, of the 13 books of Diophantus’s Arithmetica only six books survive in Greek and another four those only in Arabic.
- 4.
The earliest mention of the fifth perfect number, in the 15th century, is 33,550,336.
- 5.
Although this is true for any two rational numbers Ibn al-Bannā’ simply says “any two numbers.”
- 6.
Briefly, the argument would be that since p, q and r are rational numbers such that p 2 + q 2 = r 2, and if m, n are rationals such that p/q = m/n then, on squaring both sides of the proportion and adding 1 to the squares on each side, one sees that n 2 r 2 /q 2 = m 2 + n 2, i.e. m 2 + n 2 is also the square of a rational.
- 7.
For example, because 9 + 16 = 25, if one wants to express 36 as a sum of two rational squares one can multiply all terms by 36/25 and write (9/25) · 36 + (16/25) · 36 = 36. And each of the summands on the left is a square of a rational number.
- 8.
It was Fermat who discovered, and proved, that a whole number greater than 1 can be written as the sum of two squares exactly when all primes congruent to 3 (mod 4) in its prime factorization appear to an even power.
- 9.
Nicomachos, in his Introduction to Arithmetic, referred to these as ‘polygonal’ numbers. He also stated—relative to our earlier reference to perfect numbers—that the fourth perfect number is 8,128.
- 10.
See Djebbar, A. “L’analyse combinatoire au Maghreb.” Also the same author’s “Figurate Numbers in the Mathematical Tradition of Andalusia and the Maghrib.” (Neither of these has been published.) The present author also thanks Prof. Driss Lamrabet for sending him his doctoral thesis, containing the edited Arabic text of Ibn Mun‘im’s Fiqh al-ḥisāb.
- 11.
The consecutive whole numbers used need not begin with “1.” It was probably not until late in the twelfth century that authors in the Islamic world began claiming magical powers for these arrays when engraved on amulets, etc. Prior to that they had always been simply mathematically interesting subjects of study and were referred to as ‘harmonious arrangements of numbers.’
- 12.
The number of rows of a magic square is called its order.
- 13.
Prof. Sesiano (1988) published an edition and French translation of Abū al-Wafā’’s treatise, based on the text of the only known manuscript, which is in the Süleymaniye Library in Istanbul. The author has used this publication and Prof. Sesiano’s insights into the work in the present exposition.
- 14.
This is because for any square array of four different whole numbers to be magic the sum of any two of these integers must be the same. And that is impossible.
- 15.
Since there is no magic square of order 2 there cannot be a bordered square of order 4.
- 16.
See Sesiano, p. 121.
- 17.
In a footnote Sesiano gives the following proof. If M is the magic constant for the square and we denote by c the number in the central cell then c is the middle number in both the middle row and middle column as well as in both diagonals. Together these contain all elements of the square, other than c, once, and c four times. Hence, we may express the sum of the elements in the square as 4(M − c) + c. But this sum is also equal to the sum of all the numbers in the square, i.e. 3M. Hence, 4(M − c) + c = 3M. Since M = 15 it follows that c = 5.
- 18.
In the remainder of this argument we shall often appeal (explicitly or tacitly) to the reader’s sense of symmetry. Abū al-Wafā’ makes no reference to this concept, but—fairly clearly—is depending on it.
- 19.
This is because the sum of the other two elements is even and the sum of all three is 15.
- 20.
As Abū al-Wafā’ puts it, the difference of the two (consecutive odd/even) squares is equal to twice the sum of their sides.
- 21.
Sesiano, p. 128.
- 22.
It is worth noting that Abū al-Wafā’ wrote a book on the techniques of mental arithmetic used by merchants and government officials. And, as the reader will see from the example we work out, there is an easy procedure, with nothing involving more than additions and subtractions of small numbers.
- 23.
We do not indicate the numbers in the central 3 x 3 square, since they do not enter into the following argument.
- 24.
This is one of several results that Plutarch attributes to Hipparchos. No one knew quite what to make of this particular number until 1994 when David Hough noticed that this number is what is known as the tenth Schröder Number, named for the logician, E. Schröder. These numbers count the number of ways in which a product of ten symbols can be bracketed. For example, the product of a,b,c can be bracketed in exactly three ways—((a)(b)(c)), (((a)(b))(c)) and ((a)((b)(c)))—and the product of four symbols can be bracketed in eleven ways.
- 25.
For example, the distance of n − 1 from n is 2, the distance of n − 2 is 3, etc. So the count begins with n and ends when, working downwards, one has counted to k.
- 26.
However, taken literally his rule would lead to the (equal) expression 28 · 27 · 26 · 5. Perhaps he felt the student might want to have as few of the successive factors as possible larger than 20.
- 27.
It is to be understood that different letters may each be repeated a different number of times.
- 28.
The sukūn is a sign that looks like a very small ‘o’ and is written over a consonant to indicate that no vowel follows that consonant. For example, the Arabic word for ‘key”, which is miftāḥ, would have a sukūn over the ‘f’.’ Two consecutive consonants cannot each have a sukūn, and a word cannot begin with a letter that has a sukūn over it.
- 29.
There will be three threads because the word has three letters. There will be three colors because no letter of the word repeats.
- 30.
Ibn Mun‘im uses this term to refer to an unordered list of letters, with repetitions allowed.
- 31.
I thank Prof. A. Djebbar for sending me the Arabic text of this part of the work and, also, Prof. M. Bagheri for his help in understanding the treatise while we were waiting for a delayed flight from Tehran to Isfahan.
Bibliography
Djebbar, A. 1981. “Ensignment et recherche mathématiques dans le Maghreb des XIIIe – XIVe siècles (étude partielle).” Publications mathématiques d’Orsay, no. 81-02.
Lamrabet, D., ed. 2005. Fiqh al-Ḥisāb (The science of arithmetic). By Aḥmad Ibn-Mun‘im., Ph.D. Thesis, University of Rabat: Faculté des sciences de l’éducation.
Sesiano, J. 1988. “Le traité d’ Abū’l-Wafā’ sur les carrés magiques”. Zeitschrift für Geschichte der arabisch-islamischen Wissenschaften 12: 121–244.
Sesiano, J. 1982. Books IV to VII of Diophantus’ Arithmetica in the Arabic Translation Attributed to Qusṭā ibn Lūqā. New York, Heidelberg, Berlin: Springer.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media New York
About this chapter
Cite this chapter
Berggren, J.L. (2016). Number Theory and Combinatorics in the Islamic World. In: Episodes in the Mathematics of Medieval Islam. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3780-6_7
Download citation
DOI: https://doi.org/10.1007/978-1-4939-3780-6_7
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-3778-3
Online ISBN: 978-1-4939-3780-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)