Avicenna and Number Theory

Abstract

Among the four mathematical treatises that Avicenna takes care to place within his philosophical encyclopaedia (al-Shifā’), the one he devotes to arithmetic is undoubtedly the most singular. Contrary to the treatise on geometry, which differs little from its Euclidean model, the philosopher takes as his point of departure the treatise of Nicomachus of Gerase, but modifies its spirit to incorporate results from the many disciplines which were dealing with numbers: Euclidean Theory of numbers, Nicomachean Aritmāṭīqī, Indian reckoning, Ḥisāb, Algebra, etc. We would like to show how Avicenna, taking note of the changes in the mathematics of his time and led by a philosophical questioning about the nature of the disciplines, proposes here one of the very few texts that gives to Theory of numbers a synthetic image, gathering the themes of mathematical research for the next centuries.

3.1 Introduction

In 1981, to mark Avicenna’s millennium, Roshdi Rashed gave a lecture, published some time later under the title “Mathématiques et philosophie chez Avicenne.”¹ He started by noting that the links between mathematics and philosophy had always been strong since the beginning of Hellenistic philosophy, and that Avicenna’s direct predecessors, al-Fārābī and al-Kindī , had composed several treatises devoted exclusively to mathematics. However, in both al-Fārābī and in al-Kindī , he remarked, these writings were somewhat separate from the philosophical presentation. Yet, this was not so for Avicenna, since he conceived his mathematical treatises as an integral part of his philosophical encyclopaedia, Kitāb al-Shifā’. To understand this fact, which had never before been highlighted, by historians of science nor by historians of philosophy, Roshdi Rashed intended to refer to these mathematical writings, limiting himself, not without reason, to the sole treatise on arithmetic from the Shifā’.

From this point of view, numbers clearly offered a privileged field of study. This is perhaps due to the fact that compared to other domains, the Aristotelian conception of the classification of science, broadly adopted by Avicenna, seems to go against mathematical practice itself, since studies on numbers fell, at the time, under several related but distinct disciplines:

• Euclidean Number theory, as presented in books VII to IX of the Elements;

• What is commonly designated by the term al-Arithmāṭīqī, which refers to the neo-Pythagorean tradition in general and particularly to Nicomachus of Gerasa’s book (translated in the 9th century by Thābit ibn Qurra );²

• Integer Diophantine analysis, present in the work of mathematicians of the 10th century like al-Khāzin ;³

• Ḥisāb (“calculation”), a sometimes-eclectic combination of calculation procedures or problem solving and results on numbers;⁴

• Algebra;

• Indian reckoning, a reasoned set of calculation procedures related to the decimal place-value system.

Reverting to certain specificities of these disciplines, Roshdi Rashed showed in particular how Avicenna made algebra and Indian reckoning “secondary parts (…) of the <science> of numbers,”⁵ thus developing, following al-Fārābī , but in a slightly different way, “a non-Aristotelian area within a classification whose bias remains Aristotelian.”⁶

How then can we describe this science of numbers that Avicenna seems to have conceived? In order to answer this question, I would like to return to the Shifā’s treatise on arithmetic, thus following in Roshdi Rashed’s footsteps: not to question his conclusions in any way, on the contrary, but merely to clarify a number of points he has not had the leisure to truly address or detail.

3.2 Euclid and Nicomachus

The mathematical part of the Shifā’, as we know, consists of four treatises: geometry, arithmetic, astronomy, and music; it therefore represents, in a fairly classical way, the quadrivium disciplines. The first treatise of the series, that on geometry, is an abridgment of all thirteen books of Euclid’s Elements and the two books attributed to Hypsicles usually associated with them. Here, Avicenna scrupulously follows the sequencing of his model and although the text does not lack interest, it is therefore in no way an original composition. Due to his decision to conform to Euclid’s work, Euclidean arithmetic from books VII to IX is presented as early on as that first treatise.

The second treatise, dedicated to arithmetic and of particular interest to us, appears to follow a similar pattern in its relation with a Hellenistic model. This is the representation given in the 19th century by the eminent historian Franz Woepcke:

It is divided into four books. It is a kind of paraphrase of Nicomachean arithmetic, and the whole has little value as an original work. It is rather curious that I did not notice a single mention of Nicomachus by name throughout the course of the treatise, although Avicenna mentions Euclid’s Elements, to which he refers, and the Pythagoreans.⁷

Woepcke is undoubtedly correct on several points. The reference to Nicomachus, though implied, is necessarily present from the first: the title (al-Arithmāṭīqī) refers directly to the neo-Pythagorean⁸ tradition and, as we shall see, the general structure is the same. Secondly, it is true that compared to the work of Avicenna’s predecessors and contemporaries, the treatise doesn’t really offer new results. Finally, the comment on the Elements is particularly good since we will show shortly that the text is based, from the outset, on Euclidean arithmetic discussed in the previous treatise.

However, we believe, the treatise can’t be reduced to a paraphrasing of Nicomachus’s text. Even if the reference is obvious, we are dealing with an original composition, which offers new perspectives and incorporates new findings unknown within the strict framework of Nicomachean arithmetic.

Avicenna’s choice to formally distinguish Euclidean and neo-Pythagorean traditions should therefore not suggest that these traditions are treated symmetrically in the Shifā’. Neither does it imply that the distinction goes any further than this formal framework, the intention remaining in both cases the study of integers. In his article, Roshdi Rashed reminds us that for 10th century scientists, the gap between the two traditions was reduced to “a distinction of methods and norms of rationality.”⁹ Thus, citing Ibn al-Haytham :

Properties of numbers are shown in two ways: the first is by induction, since if we take the numbers one by one and if we distinguish between them, we find by distinguishing and considering all their properties, and to find the number in this way is called al-arithmāṭiqī. This is shown in a book on al-arithmāṭiqī [Nicomachus of Gerasa] . The other way of showing properties of numbers proceeds by proofs and deductions. All the properties of the number grasped by proofs are contained in these three books [of Euclid ] or those which refer to them.¹⁰

Avicenna’s introduction to his al-Arithmāṭiqī treatise leaves us in no doubt on how he conceived the relationship between this work and the Euclidean treatise. He starts by pointing out that the Elements (al-Usṭuqsāt) provides the science of numbers (‘ilm al-‘adad) with most of its foundations (uṣūl), adding that “it is possible to transpose to the number many geometric propositions dealing with multiplication, division and ratios.”¹¹ Then, he mentions the quiddity (māhiyya) of a number, as if to indicate what constitutes the subject of his book, contenting himself to refer to the treatise on Categories. Finally, he reminds the reader that a fair share of important topics have been covered in the Elements, such as odd, even, prime and composite numbers, even-times even, even-times odd and even-times-even-times odd, but also perfect, deficient and abundant numbers. Therefore, he deems it not necessary to go over them again.¹²

Thus, it is as if the philosopher’s intention was not so much to create a summary of Nicomachus’s Arithmetic, as to complete the Euclidian’s work in order to draw the boundaries for a science of numbers placed directly in its wake. If the topics addressed lead to a certain closeness to the neo-Pythagorean treatise (and, how could it be otherwise?), the project seems however much broader in its principle, and consequently capable of incorporating results from different sources. This is at least what we intend to show by taking up his work from its table of contents. The four books that make up the work are therein entitled:

• 1. Properties of a number;

• 2. States of a number in terms of its relationship to others;

• 3. States of a number in terms of its composition of units;

• 4. The ten proportions.

Although Nicomachus’s treatise is composed of only two books, we have found the same general progression therein. Nonetheless, the similarities affect mostly the last three points (more precisely: relationships between numbers, figurate numbers and proportions).¹³ As Avicenna’s first book, the most voluminous, is structurally very different from what we find in its Hellenistic predecessor, we will concentrate on this book in particular. It is easy to distinguish three parts:¹⁴

• 1. a study of the natural order and the succession of integers;

• 2. a first partition of integers: odd and even;

• 3. a second partition of integers: perfect, deficient and abundant numbers.

3.3 The Succession of Integers

Avicenna’s first proposition, introduced by him as “the first and most renowned” property of numbers, also coincides with Nicomachus’s first strictly mathematical proposition:¹⁵

Any number is half the <sum> of the two numbers adjacent to it, i.e. two numbers bordering its sides <at an equal distance> whether higher or lower. Example: five is half of six plus four, half of seven plus three, half of eight plus two and half of one plus nine.¹⁶

From there follows a number of outcomes, such as “the square of any number is equal to the product of its two close adjacents plus one”, i.e.:

$$ n^{2} = (n + 1)(n - 1) + 1, $$

And generalizing:

$$ n^{2} = \left( {n + a} \right)\left( {n - a} \right) + a^{2} . $$

Next comes the notion of distance between two numbers, which does not exactly cover the difference, but introduces a perception so to speak, “spatialising” the succession of numbers which is later found in a different form. Thus:

The distance of any number to its double is equal to its product by one if you don’t include it, and that plus one if it is included within the ranks.¹⁷

Expressed in terms of distances and hence often to within one unit, a number of results are then developed, such as:

$$ \begin{array}{*{20}c} {n^{2} - n = n \cdot (n - 1)} \\ {n \cdot (n + 1) - n = n^{2} } \\ {n^{3} - n = n \cdot (n + 1) \cdot (n - 1)} \\ {n^{4} - n = \left( {n^{2} + (n + 1)} \right) \cdot n \cdot (n - 1)} \\ \end{array} $$

Avicenna then returns to the study of numbers adjacent to a given number n—type $(n - a)$ and $(n + a)$—while providing other equalities, such as:

$$ \begin{aligned} 2n^{2} + 2 & = (n + 1)^{2} + (n - 1)^{2} , \\ 2n^{2} + 8 & = (n + 2)^{2} + (n - 2)^{2} , \\ 2n^{2} + 18 & = (n + 3)^{2} + (n - 3)^{2} , \\ \end{aligned} $$

Or:

$$ \begin{aligned} 2n^{2} + 4 & = (n - 1)(n - 2) + (n + 1)(n + 2), \\ 2n^{2} + 12 & = (n - 2)(n - 3) + (n + 2)(n + 3). \\ \end{aligned} $$

In both these last results, and for those expressed in terms of distance, a series of identities can be found indicating a proximity to algebra which confirm Avicenna’s use of this discipline’s lexicon: for example, his use of the expression māl māl to indicate the fourth power (square square).¹⁸

This presentation on the succession of integers continues quite naturally with the question of arithmetical sequences. Avicenna begins by announcing his intention to deal with “properties of successive numbers in natural (ṭabī‘iyya) succession,”¹⁹ in other words arithmetical sequences with common difference one, in order to then generalize to sequences with any first term and any common difference. He also gives the rules to determine the last term as well as those that provide the sum of the terms. Thus, for a sequence (u_n) with common difference d, we have:

$$ \begin{aligned} u_{n} & = u_{1} (n - 1) \cdot d \\ \sum\limits_{i = 1}^{n} {u_{i} } & = \frac{1}{2}\left( {u_{n} + u_{1} } \right) \cdot n \\ \end{aligned} $$

The results relating to the sum of various progressions were well known at the time and collections are often found in ḥisāb treatises.²⁰ What is particularly remarkable here, and much less common, is the comprehensive study of arithmetical sequences for themselves.

Also note the importance of pairs of adjacent numbers throughout this study. By considering the first n integers, Avicenna indicates that, if n is odd, numbers adjacent to the median term can be grouped two by two; the sum of any couple is, of course always the same, which naturally leads to the result giving the total sum (a quite similar reasoning gives the same result if n is even).

In this use we can see one of the reasons that led the author to introduce his first proposition as “the most renowned” property of numbers. But other purposes are also likely to be the cause of such a qualification. This particularly applies to the construction of magic squares, where pairs of numbers adjacent to the magical constant play a decisive role. An anonymous 11th century author placed this first proposition by Nicomachus and Avicenna among the preliminaries of his treatise on the subject. Furthermore the paragraph he devotes to it shows many similarities to Avicenna’s text: we find the same terminology for adjacent numbers (ḥāshiyatān) that is absent from the translation of the Hellenistic text by Thābit, and the same notion of distance between two numbers (bu‘d).²¹

Yet, the art of magic squares is not unrelated to the tradition of Nicomachus’s Arithmetic. In addition to the fact that several passages in Iamblichus tend to suggest the presence of this theme among the neo-Pythagorean school,²² we can’t fail to notice that a predecessor of Avicenna, ‘Alī ibn Aḥmad al-Anṭākī, had inserted a presentation on the construction of magic squares within his commentary on Nicomachus’s treatise.²³ Moreover, prestigious mathematicians such as Thābit ibn Qurra and Abū al-Wafā’ al-Buzjānī, also predecessors of Avicenna, had composed treatises dedicated to the very subject.

It is indeed the art of magic squares that evokes the way in which the philosopher began to exhibit a certain number of results relating to the succession of odd numbers and even-times odd numbers. Let us give an example:

If we construct a square table from odd numbers in natural succession, properties relating to this layout will appear; it is the same if we construct a triangular table. Let’s start with the square and let it be <the square of> five. We say that any cross on <this table> is either a diagonal of the <entire> figure or not. The sums of the two diagonals are equal: in relation to the diagonal, the sum of each of the two diagonals of this figure is one hundred and twenty-five; as to what is not on the diagonal <of the entire figure> , it is like the cross coming from two lines, one of which is three, fifteen, and twenty-seven, and the second seven, fifteen and twenty-three, <the sum of> each diagonal is forty-five. We find that the sum of the extremities of a line of any cross is equal to the sum of the extremities of the other line. We find that the sum of boxes of any square <constructed> from these numbers according to their succession is equal to the square of the square of the number of boxes on a side. Thus, if one constructs a square whose side is two, then its numbers are one, three, five and seven. The sum of it all is sixteen, and this corresponds to the square of the square of two.²⁴

So, we find here a different, “spatialised” approach to the study of the succession of numbers, which, although it has little relation to magic squares—the theme is never mentioned by Avicenna—does however, take up the general layout to visually express results that could otherwise be given quite differently.²⁵

3.4 Odd and Even

The developments seen above (on odds and even-times odds) are in reality part of the second section of this chapter dedicated to what Avicenna presents as “the first of the two divisions of numbers”, namely the distinction between odd and even.²⁶ The author reverts to notions explained in the Elements (quoted several times therein), and in particular to the numerous subdivisions of odd and even. He adds results relative to the succession of numbers in each of these subdivisions, once again, insisting beforehand on the importance of the role of pairs of adjacent numbers in determining the sums of arithmetical sequences. Several results are then given, like those quoted above, and the fact that the sum of a sequence of consecutive odd numbers is a square.

But properties of another nature are also explored, and other topics evoked. For example, while even-times even numbers are subject to fairly classic comments on their formation by geometric progression, and also play an important role in obtaining perfect and amicable numbers.²⁷ While the reference to perfect numbers is very succinct, since the question of their formation has been dealt with in the Elements, amicable numbers that are neither found nor mentioned in either Euclid or Nicomachus,²⁸ generate a much larger development. Besides the definition and the detailed example of the couple (224, 280), Avicenna states, if we except a superfluous condition, Thābit ibn Qurra’s theorem, which provides a general rule for their construction.²⁹

However, perfect and amicable numbers are defined by properties conferred on them by an elementary arithmetical function, the sum of an integer’s aliquot parts, i.e. the sum of its proper divisors. By calling this $\sigma_{0}$, let’s remind ourselves that:

• If $\sigma_{0} (n) < n$, n is deficient;

• If $\sigma_{0} (n) = n$, n is perfect;

• If $\sigma_{0} (n) > n$, n is abundant;

• If $\sigma_{0} (p) = q$ and $\sigma_{0} (q) = p,$ p and q are amicable.

In his treatise on amicable numbers, and in order to demonstrate his theorem, Thābit ibn Qurra had already given some properties of this function. A sign of the interest in this field of study is that Avicenna also gives results relating to this function, although he goes no further than his predecessor (for a more systematic study, we have to wait for the work of Kamāl al-Dīn al-Fārīsī).³⁰ Thus, ranked among the properties of even numbers, we find the following statements:

$$ 2n = \sigma_{0} \left( {(2n - 1)^{2} } \right){\text{if}}\,(2n - 1)\,{\text{prime}}, $$

and

$$ 2n = \sigma_{0} (2(2n - 3))\,{\text{if}}\,(2n - 3)\,{\text{prime}}. $$

3.5 Perfect, Deficient and Abundant Numbers

This leads us to the last section of this first chapter, devoted to what Avicenna considers as another division of numbers after the partition between even and odd, the one that distinguishes perfect, deficient and abundant numbers.³¹ Notice from this introduction that by straying away from the letter of Nicomachus’s treatise, which reserves these denominations exclusively for even numbers only,³² Avicenna confers this new division with greater importance. In fact, by bringing together the characteristics and properties of these numbers, which are quite numerous and often absent from neo-Pythagorean treatises, Avicenna is demonstrating the vitality of this field of research at this time.

These results are all given without proofs; some of them were conjectures, of which only a few have been found to be false. Seeming to consider that all perfect numbers are Euclidean, he starts by affirming that all perfect numbers are always even.³³ He pursues with the assertion stated by Nicomachus which, although faulty was regularly referred to until the 16th century, whereby a perfect number can be found in each interval between 1, 10, 100, 1000, 10,000, and so on with the following powers of 10; al-Baghdādī, one of his contemporaries, pointed out the erroneous character of this assertion.³⁴ He then notes that the units digit of a perfect number is always six or eight—properties also announced by al-Baghdādī,³⁵ which is always true for Euclidean perfect numbers.

Next, come a set of properties on these Euclidean perfect numbers, like the fact that one obtains a square if one multiplies them by eight and adds one,³⁶ then a process to obtain an abundant number from a perfect one which reveals another property of the function $\sigma_{0}$:

p a perfect number and q a prime number so that p and q are prime to one another; then $pq$ is abundant and:

$$ \sigma_{0} (pq) = pq + 2p $$

This result provides the first step in the demonstration of a more general theorem, stipulating that all multiples of a perfect number are abundant.

Avicenna continues with a series of criteria for whether a number is deficient or abundant, these include for example: prime numbers are all deficient; all even-times even are deficient by a unit; all multiples of six are abundant.³⁷ He ends by affirming in substance that all odd numbers are deficient unless they are composed of four consecutive odd numbers, giving the case of 945 which, he assures, is the first abundant odd. This result, also given by al-Baghdādī, was for a long time attributed to Bachet de Méziriac.³⁸

So ends the first chapter of Avicenna’s treatise. One may observe how remote, both in mind and content, it is from Nicomachus’s work.

3.6 Congruencies

The remaining three chapters, while shorter, are unquestionably far more related to the neo-Pythagorean treatise. However, it would be wrong to assume they do not contain results from other sources. Particularly the chapter on figurate numbers, which deals with a subject that experiences further developments over the following centuries.³⁹ However in this chapter, among the properties of squares and cubes, there are results on congruencies, which are particularly noteworthy. Quoting Avicenna:

Know that <the number of> units of the square number will always be either one, four, five, six or nine. If it is one, the units on its side are nine or one; if it’s four, then eight or two; if five, then five; if six, either six or four; and if it is nine, then three or seven.⁴⁰

In other words:

$$ \begin{aligned} x^{2} & \equiv 1 \,(\bmod \,10) \Leftrightarrow \left\{ {\begin{array}{*{20}c} {x \equiv 1\,(\bmod \, 10)} \\ {x \equiv 9\,(\bmod \, 10)} \\ \end{array} } \right. \\ x^{2} & \equiv 4\,(\bmod \,10) \Leftrightarrow \left\{ {\begin{array}{*{20}c} {x \equiv 8\,(\bmod \,10)} \\ {x \equiv 2\,(\bmod \, 10)} \\ \end{array} } \right. \\ x^{2} & \equiv 5\,(\bmod \,10) \Leftrightarrow x \equiv 5\,(\bmod \, 10) \\ x^{2} & \equiv 6\,(\bmod \, 10) \Leftrightarrow \left\{ {\begin{array}{*{20}c} {x \equiv 6\,(\bmod \,10)} \\ {x \equiv 4\,(\bmod \,10)} \\ \end{array} } \right. \\ x^{2} & \equiv 9\,(\bmod \, 10) \Leftrightarrow \left\{ {\begin{array}{*{20}c} {x \equiv 3\,(\bmod \,10)} \\ {x \equiv 7\,(\bmod \,10)} \\ \end{array} } \right. \\ \end{aligned} $$

Then immediately afterwards, in the same style:

Proof of the square in the Indian method: it must imperatively be one, four, seven or nine. For one, it’s either one or eight; for four, either two or seven; for seven, four or five; and for nine, it’s three, six or nine.⁴¹

The “Indian method”, in this case the famous divisibility rule for 9, is often exhibited in treatises on Indian reckoning and proves itself very useful for checking procedures where intermediate results are deleted. Thus, if we use the language of modern mathematics, Avicenna claims in substance that:

$$ \begin{aligned} x^{2} & \equiv 1\,(\bmod \,9) \Leftrightarrow \left\{ {\begin{array}{*{20}c} {x \equiv 1\,(\bmod \, 9)} \\ {x \equiv 8\,(\bmod \,9)} \\ \end{array} } \right. \\ x^{2} & \equiv 4\,(\bmod \,9) \Leftrightarrow \left\{ {\begin{array}{*{20}c} {x \equiv 2\,(\bmod \, 9)} \\ {x \equiv 7\,(\bmod \,9)} \\ \end{array} } \right. \\ x^{2} & \equiv 7\,(\bmod \,9) \Leftrightarrow \left\{ {\begin{array}{*{20}c} {x \equiv 4\,(\bmod \, 9)} \\ {x \equiv 5\,(\bmod \, 9)} \\ \end{array} } \right. \\ x^{2} & \equiv 0\,(\bmod \,9) \Leftrightarrow \left\{ {\begin{array}{*{20}c} {x \equiv 3\,(\bmod \,9)} \\ {x \equiv 6\,(\bmod \,9)} \\ {x \equiv 0\,(\bmod \,9)} \\ \end{array} } \right. \\ \end{aligned} $$

The consecutive nature of these results (modulo 10 then modulo 9) suggests the very idea of congruence, common to both sets, brought out here by Avicenna, in a way that perhaps had never been so clear.

3.7 Conclusion

In his article on Avicenna, Roshdi Rashed identifies the gap between the philosopher’s views and the neo-Pythagorean tradition, in terms of both the methods used, and the standards of rationality. He noted in particular the exclusion of “all ontological and cosmological considerations bearing on the notion of a number” from his al-Arithmāṭiqī, leaving only “the philosophical aim common to all branches of philosophy—theoretical or practical—that is the perfection of the soul”.⁴² Avicenna wrote explicitly on the subject:

It is usual, among those dealing with the art of numbers, to mention in this and similar places, developments foreign to this art and, more than that, foreign to the use of those who proceed by demonstration, and closer to the discourses of the orators and the poets. We must renounce this.⁴³

Obviously, here we are far removed from Hellenistic arithmetic, and Avicenna is drawing an entirely new landscape. The intention of the text, which could be considered as being to complement the Elements by synthetically drawing new contours for the science of numbers—at least as he conceives it—seems to us in fact to be two-fold.

Firstly, to collect and classify certain results on numbers and that can have many different origins: arithmetical sequences, odd and even numbers, aliquots parts, amicable, perfect, abundant, deficient and figurate numbers, congruencies, proportions, etc. The origin could of course be Nicomachus’s treatise, which to a certain degree provides the general framework, however, we have alternately come across many other disciplines being solicited, such as algebra, Indian reckoning, ḥisāb, and even the art of magic squares. Avicenna may well recall the recent results of his contemporaries and immediate predecessors, but he also focuses, as few had before him, on the succession of numbers and, the natural order of integers and arithmetic sequences, as well as providing advances on congruencies, and raising the partition between perfect, deficient and abundant numbers to the same rank as the distinction between odd and even.

Secondly, to exclude from number theory all that could be foreign to the knowledge of integers alone, whether it relates to the numbers being rational or irrational, or to the clarifications of calculation procedures or problem solving. This is how Avicenna explains his project at the end of the treatise:

This is what we said in the science of Arithmāṭiqī. We left some cases whose mention in this place we considered foreign to the rule of this art. What remains in the science of al-ḥisab is suitable for the use and the determination <of numbers> . Finally, what is left in the practice is like algebra and al-muqābala, Indian addition and separation, and what is similar. But it is better to mention these among the secondary parts.⁴⁴

All these disciplines—algebra, Indian reckoning, and ḥisāb—thus appear endowed with an instrumental character, which seems to oppose the concept of the philosopher. Note that there is neither any mention nor examples of integer Diophantine analysis. It is possible that Avicenna had not grasped all the issues involved in this still relatively new discipline. In a passage from the Logic section of the Shifā’, where he evokes the first case of Fermat’s conjecture—on whether the sum of two cubes is a cube—he seems to think that the problem falls within ḥisāb⁴⁵, which would render it foreign to his topic.

The number theory outlined by Avicenna is of course not specific to him: features date as far back as Thābit ibn Qurra’s memoir on amicable numbers, which tackles a neo-Pythagorean subject in a deliberately Euclidean style. Nevertheless, his treatise remains remarkable. Not so much by its content that one can often find elsewhere, nor by his methods (few of which are detailed), but by his globalizing aims and sharpness of the boundaries he draws. In this, this text differs from most mathematical writings of the time, whether pamphlets devoted to more circumscribe topics—like Thābit’s memoir on amicable numbers—or much broader texts like al-Baghdādī’s al-Takmila fī al-ḥisāb, where properties of integers coexist with procedures from Indian reckoning and the arithmetic of irrational numbers.

Noting the changes taking place within the mathematics of his time, and driven by an enquiring philosophical mind which leads him to question the nature of the various disciplines, Avicenna thus proposes one of the very few texts to give the from then on reunited number theory a synthetic image bringing together, with the exception of integer Diophantine analysis, themes for future research.