# Indeterminate linear problems from Asia to Europe

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## 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.

## Keywords

Diophantine equations Simultaneous linear congruences Zhang Qiujian Sun-Tzu 100 fowls problem Ta-yen Kuṭṭakā Chinese remainder problem## Notes

## References

- 1.Āryabhaṭīya of Āryabhaṭa/critically edited with introd., English translation, notes, comments, and indexes by Kripa Shankar Shukla, in collaboration with K. V. Sarma. The Indian National Science Academy, New Delhi (1976)Google Scholar
- 2.Boncompagni, B.: Scritti di Leonardo Pisano matematico del secolo decimo terzo. 2 vols.: I. Il Liber abaci pubblicato secondo la lezione del Codice Magliabechiano C. 1., 2616, Badia Fiorentina, n. 73 da B.Boncompagni ; II. La pratica geometriae. Opuscoli: Flos, le Questiones avium e il Liber quadratorum, pubblicati da B. Boncompagni. Boncompagni Tipografia delle Scienze matematiche, Rome (1857–1862)Google Scholar
- 3.Chemla, K.C.: East Asian mathematics. In: Encyclopædia Britannica. Encyclopædia Britannica, Inc. (2011), https://www.britannica.com/science/East-Asian-mathematics. Accessed 29 Nov 2018
- 4.Caianiello, E.: Des monnaies et des oiseaux dans l’œuvre de Léonard de Pise. Revue de Numismatique
**6**, 151–166 (2011)CrossRefGoogle Scholar - 5.Colebrooke, H. T.: Algebra with arithmetic and mensuration from the Sanscrit of Brahmagupta and Bh'ascara, p. 378 (1973, Reprint of 1817 ed.)Google Scholar
- 6.Datta, B., Singh, A.N.: History of Hindu Mathematics: A Source Book. Asia Publishing House, Calcutta (1935–1938)Google Scholar
- 7.Daumas, D., Guillemot, M., Keller, O., Mizrahi, R., Spiesser, M.: Le théorème des restes chinois. Textes, commentaires et activités pour l’arithmétique au lycée. In: Le problème des restes chinois: Questions sur ses origines, IREM, Paris (2011). http://culturemath.ens.fr/materiaux/irem-toulouse11/theoreme-restes-chinois-index.html. Accessed 29 Nov 2018
- 8.Djebbar, A.: Les transactions dans les mathématiques arabes: classification, résolution et circulation. In: Actes du Colloque International “Commerce et Mathématiques du Moyen Age à la Renaissance, autour de la Méditerranée”, pp. 327–344. Editions du C.I.H.S.O., Toulouse (2001)Google Scholar
- 9.Folkerts, M.: Die älteste mathematische Aufgabensammlung in lateinischer Sprache: Die Alkuin zugeschriebenen Propositiones ad Acuendos Iuvenes. Springer, Berlin (1977)zbMATHGoogle Scholar
- 10.Hayashi, T.: Indian mathematics. In: Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, vol. 1, pp.118–130. Routledge, London (1994)Google Scholar
- 11.Hayashi, T.: The Bakhshālī manuscript: an ancient Indian mathematical treatise. Egbert Forsten, Groningen (1995)zbMATHGoogle Scholar
- 12.Hayashi, T.: Mahāvirā. In: Encyclopædia Britannica. Encyclopædia Britannica, Inc (2017). https://www.britannica.com/biography/Mahavira-Indian-mathematician. Accessed 29 Nov 2018
- 13.Kangshen, S., Crossley, J.N., Wah-Cheung Lung, A., Liu, H.: The Nine Chapters on Mathematical Art, Companion and Commentary. Oxford University Press, Oxford (1999)zbMATHGoogle Scholar
- 14.Libbrecht, U.: Chinese Mathematics in the thirteenth century. The MIT Press, Cambridge (1973)zbMATHGoogle Scholar
- 15.Mahāvirācarya: The Ganita-Sāra-Sangraha, Rangācārya, M.A. (Engl. transl. and ed.). Madras (1912)Google Scholar
- 16.Martzloff, J.C.: Histoire des mathématiques chinoises. Masson, Paris (1987)zbMATHGoogle Scholar
- 17.Migne, I.M.: Propositiones Alcuini doctoris Carolo Magni Imperatori ad acuendos juvenes. In: Alcuini Opera Omnia (Patrologiae cursus completus), t. 101, vol. 3, Paris (1851)Google Scholar
- 18.Needham, J.: Science and Civilisation in China. Vol. 3. Mathematics and the Sciences of the Heavens and Earth. Cambridge University Press, Cambridge (1959)Google Scholar
- 19.Rashed, R., Morelon, R. (eds.): Encyclopedia of the History of the Arabic Science. In: Mathematics and the physical science, vol. 2. Routledge, London, New York (1996). https://archive.org/details/RoshdiRasheded.EncyclopediaOfTheHistoryOfArabicScienceVol.3Routledge1996. Accessed 29 Nov 2018Google Scholar
- 20.Saidan, A.S.: The Arithmetic of Al-Uqlīdisī. The Story of Hindu-Arabic Arithmetic as told in Kitāb al-Fuṣūl fī al-Ḥisāb al-Hindī. Springer, Berlin (1978)zbMATHGoogle Scholar
- 21.Sarasvati, S.S.P., Jyotishmati, U.: The Bakhshālī Manuscript: An Ancient Treatise of Indian Arithmetic. Dr. Ratna Kumari Svadhyaya Sansthan, Allahabad (1979)Google Scholar
- 22.Shukla, K.S.: The Pātīganita of Sridharacarya with an Ancient Sanskrit Commentary. Lucknow University, Lucknow (1959)Google Scholar
- 23.Smith, D.E.: History of Mathematics, vol. 2. Dover, Mineola (1958)Google Scholar
- 24.Souissi, M.: Le talkhis d’Ibn al-Bannā. In: Chabert, J.-L. et al (eds.) Histoire d’algorithmes. Du caillou à la puce. Belin, Paris (1991)Google Scholar
- 25.Suter, H.: Das Buch der Seltenheiten der Rechenkunst von Abū Kāmīl el Misri. Bibliotheca Mathematica
**3**11, 102 (1910/1911)Google Scholar - 26.Tabak, J.: Algebra: Sets, Symbols, and the Language of Thought. Infobase Publishing, New York (2009)Google Scholar
- 27.Vanhée, L.: Les cent volailles ou l’analyse indéterminée en Chine. T’oung Pao
**14**, 435–450 (1913)CrossRefGoogle Scholar - 28.Vinogradov, I.M.: Elements of Theory of Numbers. Dover, Mineola (1954)Google Scholar