Indeterminate linear problems from Asia to Europe

Abstract

In the Chinese tradition, linear Diophantine problems can be traced back to two main categories. The first group includes the problems consisting of 2-equation systems with integer coefficients in n unknowns with $n> 2$ and integer solutions, such as: $x_1 + x_2 + \dots + x_n = p$, $b_1 x_1 + b_2 x_2+ \dots + b_n x_n = q$. The so-called “100 fowls problem” belongs to this group. The oldest statement can apparently be found in the Zhang Qiujian Suanjing (The Computation Classic of Zhang Qiujian), towards the second half of the 5th century AD (468–486). In the second category there are problems that can be represented through simultaneous linear congruences, which are generally formulated as follows: find a number x that, divided by $m_1$, $m_2$, $m_3, \ldots , m_i$, give as remainders $r_1$, $r_2$, $r_3, \ldots , r_i$. This kind of problem is currently called “Chinese (remainder) problem”. The oldest formulation is attributed to Master Sun Tzu (between 280 and 473). We shall present the different statements of the two problems in their circulation from Asia to Europe and the main proof procedures.

Appendices

Appendix 1: About packet counting

In the classic method of consulting the oracle, which the I Ching (Book of Changes) describes ..., one of the fifty stalks or rods is set aside before the forty-nine are divided into two random heaps symbolising the Yin and the Yang. It was very natural, therefore, that mathematicians, seeking for remainders of one by continued divisions, should have remembered this.

Thus writes the eminent Chinese civilisation scholar Joseph Needham in Science and Civilisation in China [18, pp. 119–120, note j]. He continues:

Other “remainders” follow, for the divining process continues by throwing out the residual rods when each successive heap is counted through by fours. In China, therefore, indeterminate analysis was connected with, if not actually derived from, an ancient method of divination using yarrow stalks. An association of these with the counting-rods at some time may perhaps be presumed.

Therefore, in the I Ching, at least in its original version, the remainder of the division by 4 has a special importance for divination, but subsequently also the remainders by 3, by 2, etc. While the modulo can change, what matters is the search for the remainder. This is the case in many counting-based divination practices of traditional societies. In Quechua-speaking Andean peoples, for instance, when sowing, a cupful is extracted from a sack of wheat, the extracted seeds are thrown on the ground, and then they are divided in two heaps. If the remainder is zero, it is a favourable omen. The remainder in the count by two is widespread, but is not the only one; the remainders different from seven in the count by ten are of good omen among the Kikuyu population of Kenya [7]. There are innumerable examples. Another interpretation of the origin of Sun Tzu’s problem is given by the 13th-century Chinese mathematician, Yang Hui, who wrote in his Yang Hui suanfa (The Computational Methods of Yang Hui) (c. 1275) that Master Sun Tzu’s problem was usually called “the secret method of Prince Ch’in to count the soldiers”. Yarrow stalks or soldiers, it is always about counting. On the other hand, as many ethnographic data confirm, the practice of counting by packets has been ingrained since the beginning in the activity of counting and computation systems. So, even though it is controversial (see Libbrecht [14, p. 367 ff.]), why not give credit to this hypothesis regarding the origin of indeterminate analysis in China?

Appendix 2: The pulveriser (kuṭṭakā)

1.1 First example

$$\begin{aligned} 5y \equiv 3\; (\bmod \;{17})\ \leftrightarrow \ 5y = 3+17x\ \leftrightarrow \ y=\frac{3+17x}{5} \end{aligned}$$

(4)

Let us apply the Euclidean algorithm to the dividend and the divisor:

$$\begin{aligned} 17&= 5\cdot \underline{3} + 2\\ 5&= 2\cdot \underline{2} + 1. \end{aligned}$$

Let us count the number of quotients: it is even.

Two columns are arranged, with quotients and remainders:

$$\begin{array}{*{20}l} 3 & {\quad 2} \\ 2 & {\quad 1.} \\ \end{array}$$

In the quotient column, two positive integer numbers w and z that satisfy the following equation are added:

$$\begin{aligned} z=\frac{1\cdot w-3}{2} \end{aligned}$$

w must be a positive integer whose product by the last remainder, minus the constant term of (4), is divisible by the last divisor (or the next-to-last remainder) and is obtained by trial and error; z is the quotient of this division. (If the number of quotients were odd the constant term would be added.) So:

$$\begin{aligned} \begin{array}{r} 3\\ 2\\ w=5\\ z=1 \end{array} \end{aligned}$$

Alongside the column of quotients we construct two other columns whose terms are obtained by successively multiplying w by the number occupying the upper position and adding the result to the number of the lower position: $5\cdot 2 + 1 = 11$. The number thus obtained is in turn multiplied in the next column by the number immediately above it, and the product added to the number immediately under it. So:

$$\begin{array}{*{20}l} 3 \hfill & {\quad 3} \hfill & {\quad 38} \hfill \\ 2 \hfill & {\quad 11} \hfill & {\quad 11} \hfill \\ 5 \hfill & {\quad 5} \hfill & {} \hfill \\ 1 \hfill & {} \hfill & {} \hfill \\ \end{array} {\text{ }}$$

The two values in the last column give the solutions to the equation: $x = 11$ and $y = 38$. Now we divide $y = 38$ by 17, and 11 by 5. The smallest solutions $x_0$, $y_0$ are given by the congruence classes modulo, respectively, the coefficients b ($= 5$) and a ($= 17$) of equation (4):

$$\begin{aligned} x \equiv x_0\; (\bmod \;{5}) \hbox { and } y \equiv y_0\; (\bmod \;{17}). \end{aligned}$$

By replacing the numerical values we obtain:

$$\begin{aligned} 11 \equiv 1\; (\bmod \;{5}) \hbox { and } 38 \equiv 4\; (\bmod \;{17}); \end{aligned}$$

thus, $x_0 = 1$ and $y_0 = 4$ are the smallest solutions of (4).

1.2 Second example

$$\begin{aligned} \begin{array}{c} \begin{array}{c} 137x + 10 = 60y\\ \\ 137 = 60\cdot 2 + 17\\ 60 = 17\cdot 3 + 9\\ 17 = 9\cdot 1 + 8\\ 9 = 8\cdot 1 + 1 \end{array}\\ \\ \begin{array}{cc} \hbox {Column of quotients} &{} \hbox {Column of remainders}\\ 2&{}17\\ 3&{}9\\ 1&{}8\\ 1&{}1 \end{array} \end{array} \end{aligned}$$

Since the number of coefficients is even, we choose a multiplier w such that the product of w by the last remainder decreased by 10 is divisible by the last divisor (or next-to-last remainder) 8; $z=(w-10)/8$, hence: $w=18$ and $z=1$.

Let us construct a table, as above:

$$\begin{aligned} \begin{array}{|c|c|c|c|c|} \hline 2&{}2&{}2&{}2&{}297=y\\ \hline 3&{}3&{}3&{}130&{}130=x\\ \hline 1&{}1&{}37&{}37&{}\\ \hline 1&{}19&{}19&{}&{}\\ \hline 18&{}18&{}&{}&{}\\ \hline 1&{}&{}&{}&{}\\ \hline \end{array} \end{aligned}$$

Thus, $x=130$, $y=297$. Reasoning as in the previous example, we find $130 \equiv 10\; (\bmod \;{60})$ and $297 \equiv 23\; (\bmod \;{137})$. Hence, the smallest solutions are given by $(x_0, y_0)=(10, 23)$.

Appendix 3: Fragments of the history of Chinese mathematics (from the Han period to the 13th century)

In the early Han period (208 BC–206 AD) the two oldest mathematical texts of which we have a written testimony appear: Zhoubi Suanjing (The Zhou Dynasty Canon of Gnomonic Computations), referred to as ZBSJ, a treatise of quantitative cosmology (containing a kind of decimal numeration, applications of the Pythagorean theorem, the value 3 for the ratio of circumference to diameter), and Jiuzhang Suanshu Suanshu (The Nine Chapters on the Mathematical Art, simply called The Nine Chapters), referred to as JZSS, of uncertain date, which became, in the Chinese tradition, the classic of classics, the Chinese mathematical bible. It contains arithmetic, algebraic and geometric algorithms, presented in relation to various problems, including some which allude to the tasks of the Han dynasty administrative officials, such as measuring fields, taxation based on wheat types, the salaries of officials according to hierarchy, excavation works, etc. The mathematical contents comprise topics in arithmetic (the rule of three, proportional division, rule of simple and double false position), geometry (calculation of surfaces and volumes), algebra (linear systems with the Fangcheng rule, including negative numbers). As with every canonical work, many of the results collected in the Nine Chapters date back to earlier times and over the centuries many authors added explanations, clarified passages, provided proofs or new procedures and drafted countless commentaries. The most important is that by Liu Hui (late 3rd century AD), since it is the first text of Chinese mathematics, before the 13th century, that provides proofs in the modern sense.

From the 3rd to the 6th century AD, the mathematical procedures become more theoretical: less importance is given to the empirical verification and more to proofs. A reflection on the correctness of the procedures was begun. Liu Hui computed the value of $\pi = 155/50$, while Chu Chongzi found the value 355/113 at the end of the 5th century. Liu Hui showed an important algorithm to determine the volume of solids based both on their reconstruction through elementary components and on their decomposition into non-arbitrary, even infinitesimal, elements. Thus, the volume of the sphere is found by applying a generalised version of what came to be known in the West as “Cavalieri’s principle” and the volume of the pyramid is found through a division into infinitesimal surfaces. Many of these topics were then developed in commentaries on the JZSS. Science and technology made a leap forward: in cartography, Pei Xiu introduced the division into rectangular grids (end of the 3rd century), Yu Xi (307–338) discovered the precession of the equinoxes, Chu Chongzhi invented machines (late 5th century).

Starting in the time of the Sui dynasty (588–618) and throughout the Tang empire (618–907), mathematics, even though it was considered a marginal subject with respect to the humanistic education of the Confucian scholars, was officially taught in the guozzxue (School for the Sons of the State) based on a collection of mathematical texts, assembled during the 7th century AD by a group of scholars under the guidance of the mathematician and imperial astronomer Li Chunfeng [3]. This collection contained contemporary and ancient mathematical writings and is called The Ten Computational Canons (SJSS).²¹ The level of these texts is lower than the Nine Chapters’, but new topics in number theory appeared, such as “100 fowls problem”, “Master Sun’s problem” and a procedure to extract a generalised cubic root. The official teaching of mathematics declined at the end of the Tang dynasty, then disappeared for about two centuries with the formation of the five independent kingdoms. It was re-established after the reunification of the empire under the Song dynasty, first Northern Song (960–1127), later Southern Song (1127–1279), even if in a more discontinuous and fragmentary way. After 1113 it completely disappeared from the curricula for the imperial exams until the 1887 reform.

From the 10th to the 12th century, the scarcity of written sources makes it difficult to accurately describe mathematical activity in China. From allusions and references by later authors, it seems that Xian and Liu Yi, mathematicians of the 11th century, knew the method known in the West as Ruffini-Horner’s method, Pascal’s triangle (referred to as an ancient procedure), and the resolutive procedure of magic squares.

In 1084 the Ten Canons were printed, apparently occasioned by a renewal in the studies during the 11th and 12th centuries. This may have paved the way to the blossoming of mathematical studies in the second half of the 13th century. The golden age of Chinese algebra and mathematics in general corresponds to the half-century between 1247 and 1303, with four major mathematicians at work:

• Qin Jiushao (c. 1202–1261), author of the Shushu Jiuzhang (1247) (Nine-chapter treatise of mathematics), known for having formulated the general algorithm to solve simultaneous congruences;

• Yang Hui, about whom it is known only that he lived under the Southern Song, and who worked in the area of elementary arithmetic. His books are among the few contemporary Chinese mathematics works to survive. Among these we mention the Xiangjie jiuzhang suanfa (A Detailed Analysis of the Nine Chapters on the Mathematical Procedures)(1261) and the Yang Hui suanfa (The Computational Methods of Yang Hui, c. 1275), consisting of three treatises;

• Li Zhi (1192–1279), the author of several works on polynomial algebra, even with negative exponents for the unknown;

• Zhu Shijie (late 13th century), the latest algebraist of his century, author of Szjuan Yujzan (1303) (Jade Mirror of the Four Origins), who handled complex algorithms and equations up to the 14th degree.

Qin Jiushao and Yang Hui lived in Southern China; Li Zhi and Zhu Shijie in the North. Finally, we must remember Guo Shoujing (1231–1316) who introduced spherical trigonometry. According to Martzloff [16, p. 20 ff], 13th-century Chinese mathematicians carried out very similar work, although they apparently did not have contact with each other and did not refer to each other in their texts. They shared a language whose terms were often borrowed from the esoteric language, and a way of conceiving algebraic activity that was much more abstract and free from realistic constraints than in the past, aimed at solving very complex problems with sophisticated and sometimes abstruse algorithms. The great expansion of algebra and mathematics took place in the period when China came into contact with Islamic mathematical culture through the Yuan (Mongol) dynasty (1264–1368); many topics were common but the procedures were very different.

Appendix 4: Mahāvirā

Mahāvirā, known as “Mahāvirā the master”, a Jain mathematician who lived in the 9th century AD—not to be confused with Vardhamana Mahāvirā (Mahāvirā means in Sanskrit “great hero”), born around 599 BC and founder of Jainism—was active in Karnataka, in South-Western India, and lived during the reign of Amoghavarṣa (c. 814–880). He was the author of a single work, the Ganita-Sāra-Sangraha (Compendium of the Essence of Mathematics), the first treatise exclusively about mathematics in the history of Indian mathematics. It consists of 1130 verses divided into nine chapters; the first is on terminology and the other eight concern mathematical procedures called pāṭī (“algorithms”):

• 1. Terminology;

• 2. Procedures for basic operations;

• 3. Procedures to reduce fractions to a common denominator;

• 4. Procedures for various problems;

• 5. Procedures on the rule of three;

• 6. Procedures on compounds;

• 7. Procedures for geometric computing;

• 8. Procedures on excavations;

• 9. Procedures on shadows.

The topics covered in the work include basic operations, reduction of fractions, miscellaneous problems involving rates of interest, proportional distribution, first- and second-degree determinate and indeterminate equations (with the method of the “pulveriser”), number series, combinatorial calculus applied to Sanskrit prosody, geometric calculations with plane and solid figures and shadows.

Mahāvirā made a great contribution to the terminology about the definitions of plane figures. Writing about zero and negative numbers, he states that a negative quantity does not have a square root because it is not a square. His contribution to the decomposition of a fraction in the sum of unit fractions is original. It also seems that he was the first Indian mathematician to admit two solutions for second-degree equations. We recall, among his innumerable results, the discovery of identity: $a^3 = a(a+b)(a-b) + b^2(a-b) + b^3$ and the identification of the formula:

$$\begin{aligned} C_{n,k} = \frac{n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)\cdots 2\cdot 1}. \end{aligned}$$

For more about Mahāvirā, see [10, 12, 26].