Numbers, Numbers

by FSTC Published on: 14th September 2009

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Amicable number, perfect numbers, deficient numbers, abundant numbers, studying numbers was done by many including Ibn Sina better known for work in medicine.

bannerNumber theory interested the Greeks and they studied special kinds of whole numbers (even, odd, squares, etc.) This interest continued to the end of the ancient period. Euclid thought up “perfect numbers”, which are the sum of their proper divisors, an example being 6=3+2+1, where 3, 2 and 1 are divisors. Therefore, ideas around the study of numbers had a long history in the ancient world.

The explorations in fields of Arabic mathematics started in number theory and mathematicians made great contributions building on the studies found in earlier mathematical traditions. It has been proved that the first great contribution to number theory, in particular, regarding amicable numbers, is credited to mathematicians of Arabic science.

The beginning of Arab contribution in number theory emerged with the mathematician Thabit Ibn Qurra and the theorem he expounded in Kitab al-a’dad al-mutahabba (Book of Amicable Numbers). Other mathematicians, such as Kamal al-Din Al-Farisi and Muhammad Baqir Yazdi, contributed to number theory and they obtained their results by using Thabit’s theorem. Historians showed that the study of number theory formed a continuous tradition and led to the discovery of theorems or problems usually ascribed to Western mathematicians several centuries later. For example, the appearance of Wilson’s theorem in the work of Ibn al-Haytham, Bachet’s problem of the weights in Al-Khazini, or the summation of the fourth powers of the integers 1,2,… n in the work of 10th-century mathematician Abu Saqr al-Qabisi. Although he is more widely known for his work in medicine, Ibn Sina (or Avicenna, as he is known in Europe), also provided some work on number theory.

Arabic works on Number Theory

Let us start with Ibn Sina and some of his works on the number theory. His important work entitled Alai in Persian and Kitab Al-Shifa in Arabic (Book of Healing), contains sections on arithmetic. He began a discussion, based on Greek and Indian sources, of different types of numbers (e.g. odd, even, deficient, perfect and abundant numbers) and explained different arithmetical operations, including the rule for ‘casting out nines’. Examples from this work are the explanations:

  • 6 is a perfect number since the sum of its proper devisors is 1+2+3=6,
  • 8 is deficient number since the sum of its proper devisors is 1+2+4<8,
  • 12 is abundant number since the sum of its proper devisors is 1+2+3+4+6>12,

The rule of casting out nines is:

The sum of digits of any natural number when divided by 9 produces the same remainder as when the number itself is divided by 9. For example,

1- Add the digits of the number 436 to get 13, whose digits are then added to get 4 (the remainder when divide the number by 9),

2- Add the digits of the number 659 to get 20, whose digits are then added to get 2 (the remainder when divide the number by 9),

3- The product of the two numbers(436 and 659) is 287324, add the digits to get 26, whose digits are then added to get 8 (the remainder when divide the number by 9),

So casting out nines leaves remainder of 4, 2 and 8 respectively, and since 4×2=8, the multiplication is probably correct.

Ibn Sina states two rules, the first one:

If successive odd numbers are placed in a square table, the sum of the numbers lying on the diagonal will be equal to the cube of the side; the sum of the numbers filling the square will be the fourth power of the side. Figure 1 illustrates this rule from the odd number in the square as follows:

9 7 5 3 1
19 17 15 13 11
29 27 25 23 21
39 37 35 33 31
49 47 45 43 41

Figure 1. Illustration of Ibn Sina’s rule 1.
Which is equal to the cube of the ‘side’ 53. The total number of the square is 625 = 54; the fourth power of the side. Therefore, Ibn Sina knew that the sum of successive odd numbers starting with 1 is equal to the square of the number of odd numbers being added. For example:

1+3+5+7+9+11=36, which is 62 (6 is the number of odd numbers).

Ibn Sina’s second rule is for summing a triangular array of odd numbers, that: If successive odd numbers are placed in a triangle, the sum of the numbers taken from one row equals the cube of the (row) number.

The triangular array of the odd numbers from 1 to 30 is shown in figure 2. The sum of the numbers in, say, the fourth row is 64 = (13+15+17+19), which equals to 43 (the cube of the row number 4).

5 3
11 9 7
19 17 15 13
29 27 25 23 21

Figure 2. Illustration of Ibn Sina’s rule 2

Now let’s find out more about the number theory from Thabit Ibn Qurra, the first mathematician who provided a great contribution. He discovered the formula of amicable numbers. Based on the fact that: A pair of natural numbers, M and N, are defined as amicable if each is equal to the sum of the proper devisors of the other.

Therefore, Thabit stated his formula for deriving pairs of amicable numbers as following:

Let p, q and r be distinct prime numbers given by

Where n is greater than 1, then M and N will be a pair of amicable numbers such that:


For n = 2,

Now p, q and r are all prime numbers, so


which is the smallest pair of amicable numbers.

It can be noticed that:

The proper devisors of 220 are

1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110.

The sum of which is 284. Similarly, the proper devisors of 284 are

1, 2, 4, 71, 142.

The sum of which is 220.

Jan P. Hogendijk (1985) pointed out that Thabit carried through his proof of his theorem for the case when the parameter n = 7 which indicates that Thabit knew the amicable pair 17,296 and 18,416 (the amicable numbers produced for n = 4 ).

It has been shown that other mathematicians obtained their results by using Thabit’s theorem. One of them is the mathematician Kamal al-Din Farisi who identified in the 13th century the two amicable numbers 17,296 and 18,416 in the case of n = 4 in Thabit’s theorem. Then, in the early 1600s, Muhammad Baqir Yazdi identified the two amicable numbers 9,363,584 and 9,437,056 in the case of n = 7 in Thabit’s theorem.

The discovery of the two couples of amicable numbers (17,296 and 18,416; 9,363,584 and 9,437,056) is usually attributed to Fermat and Descartes. But it has been recently shown that Fermat’s couple had been calculated by an earlier Moroccan mathematician, Ibn al-Banna (1256-1321), who calculated it at least a century earlier. This mathematical result was subsequently known to many mathematicians, as was the case for the so called “Descartes’ couple”. This argument can be realised by focusing on the means implemented for the calculation of amicable numbers which will indicate who started what.

For this situation we can return back to Al-Farisi’s work on amicable numbers as evidence. Al-Farisi was not satisfied just to give the calculation of “Fermat’s couple” but stated a complete justification for it as well. He started with n = 4, then from Thabit’s formula (above) he gets:

The first two numbers are obviously primes, and he used several propositions to show that 1151 is a prime number. To prove that Fermat’s couple is really a couple of amicable numbers, Al-Farisi proceeded as follows:

First recalling the definitions:

For a natural number the sum of its proper (excluding n) divisors is so that:


By considering all the possible proper divisors of the number 2kI where I is a prime number or a product of prime numbers we can note that the sum of all the possible divisors of 2kI is:

The can be rewritten through the following steps into a calculation with smaller numbers to work with:

This formula was used by Al-Farisi to verify the amicable number pair 17,296 and 18,416:

=15(71+1081) + 16(71) = 18416,

= 15(1+1151) +16(1) = 17296,

These works on Number Theory were a great contribution in the field of mathematics by great mathematicians and show that the Arabic scientific tradition played a pioneering role in Number Theory, as it was the case also in other areas of mathematics.


1- F.J. Swetz: From Five Fingers to Infinity; Open Court; Chicago; 1994; pp. 289-92.

2- G.G. Joseph: The Crest of the Peacock; Penguin Books; 1991.

3- J.L. Berggren: History of Mathematics in The Islamic World: The Present State of The Art. Middle East Studies Association Bulletin 19 (1985), pp. 9-33.

4- R. Rashed: The Development of Arabic Mathematics: between arithmetic and algebra. Dordrecht: Kluwer. 1994.

5. Jan P. Hogendijk, “Thabit ibn Qurra and the pair of amicable numbers 17296, 18416”, Historia Mathematica, vol. 12 (3) (1985), pp. 269-273.

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