• What is special about these numbers: 6, 28, 496 and 8128?
• What is the nature of the relationship between the following pairs of “amicable,” numbers, as they are rather exotically denoted: 220 and 284, 17296 and 18416, 9 363 584 and 9 437 056?
• Why are the numbers 2, 3, 5, 7 drawn from a special infinite series referred to as the “atoms of arithmetic?”

Carl F Gauss Stamp

These numbers, and many other kinds besides, are the subject of what Carl Friedrich Gauss (1777 – 1855) – one of the greatest Mathematicians to have ever lived – liked to call the Queen of Mathematics. An enviable position, particularly considering that many see Mathematics as the Queen of the Sciences. In specialist circles, this field is known by the more prosaic title of Number Theory.

As with so many fields of knowledge, the Greeks were likely the first people to study the properties and relationships of numbers in a systematic and rigorous manner. The first group of numbers above – 6, 28 and so on – are called perfect numbers­ and were studied by Pythagoras, especially for their supposed mystical qualities, and none other than the father of geometry, Euclid. Perfect numbers are defined as those numbers that are equal to the sum of their proper divisors ^[1]. For instance, the proper divisors of 6 are 1, 2 and 3, which sum to 6; and we also observe, 28 = 1 + 2 + 4 + 7 + 14.

Ibn al-Haytham

Al-Hasan ibn al-Haytham (d. 1040, Cairo) is most noted for his contributions in the field of optics. As was the case for many of the scholars that Islamic civilization has bequeathed, ibn al-Haytham’s intellectual curiosity motivated him to conduct explorations in a huge array of subjects, including number theory. Specifically, he attempted to characterize the set of perfect numbers, which he sets out in his unpublished work Analysis and Synthesis. Euclid had already proved in the Elements the elegant result that if 2^k – 1 is prime ^[2], then 2^k–1 .(2^k – 1 ) is a perfect number ^[3], for any integer k greater than 1. As an application of this theorem, take k = 3. Then, 2³ – 1 = 7 is a prime number and 2². (2³ – 1) = 28 is the second perfect number. The interesting question is whether all the perfect numbers can be generated by Euclid’s formula. This question is not as straightforward as it sounds; in actual fact, it still remains an open question amongst mathematicians! It is quite typical of many mathematical theorems that they go in one direction and not the other, and Euclid’s theorem on perfect numbers is an example of that. Now, Ibn al-Haytham was interested in demonstrating the converse result. He wasn’t entirely successful – we can forgive him this shortfall given that the problem has eluded some of the best mathematicians. However, there are suggestions in Analysis and Synthesis ­that he was the first to attempt the proof of the partial result that every even perfect number is of the form 2^k–1 .(2^k – 1 ) where 2^k – 1 is prime. The world had to wait seven centuries before Leonhard Euler provided a complete, correct and economical proof of the converse theorem.

Before we leave Ibn al-Haytham, we should note that he appears to be the first person to discover and use what came to be known as Wilson’s theorem, which states that if p is prime, then^[4] 1 + (p – 1)! is divisible by p. The indications are that the first proof of this theorem was produced by Lagrange in 1771 and, as remarked by O’Connor and Robertson,

…it is more than 750 years after al-Haytham before number theory surpasses this achievement of Arabic mathematics.” Arabic mathematics: forgotten brilliance?​ by J J O’Connor and E F Robertson

The kind of problems for which ibn al-Haytham employed Wilson’s theorem is illustrated in Opuscula:

To find a number such that if we divide by two, one remains; if we divide by three, one remains; if we divide by four, one remains; if we divide by five, one remains; if we divide by six, one remains; if we divide by seven, there is no remainder.”  John Wilson’s Opuscula

Ibn al-Haytham describes a general method to solve the problem above, which, for the given example, produces the solution, (7 – 1)! + 1. We can see, invoking Wilson’s theorem, that this number is divisible by the prime number 7, and that it clearly leaves the reminder 1 when divided by any of 2, 3, 4, 5 and 6. (Observe (7 – 1)! + 1 = 6.5.4.3.2 + 1.)

We now steer our journey back in time and away from Cairo, where Ibn al-Haytham carried out his most seminal work, to the previous century and peer into the celebrated House of Wisdom in Baghdad during the time of the Abbasid Caliph, al-Mu’tadid. On his travels to Harran (now in Turkey), Mohammed bin Musa bin Shakir, one of the Banu Musa brothers, who already had gained a reputation for sponsoring and personally undertaking remarkable studies in the sciences, spotted the linguistic talents and scholarly potential of Thabit bin Qurra (d. 901, Baghdad).

The round city of Baghdad in the 10th century at the time of House of Wisdom. Illustration: Jean Soutif/Science Photo Library (Source)

Thabit accompanied his future patron to Baghdad where he would complete ingenious and penetrating investigations in not only mathematics, but ” … logic, psychology, ethics, the classification of sciences, the grammar of the Syriac language, politics, the symbolism of Plato’s Republic … religion and the customs of the Sabians.” Thabit’s contribution to mathematics was summarised aptly by Rashed, he has reported:

… played an important role in preparing the way for such important discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry and non-Euclidean geometry. In astronomy, Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics.” Roshdi Rashed   more: http://www.muslimheritage.com/article/hail-queen-mathematics