Abstract
This chapter analyses Li Shanlan’s 李善蘭 (1811–1882) original attempt to disclose the Euclidean canon by adding an extra procedure for testing the primality of a number. Li is a particularly interesting figure since he worked and taught mathematics in a compartmentalized fashion: either in traditional algorithmic Chinese style or by adopting a syncretistic symbolism, integrating Chinese linguistic components into algebraic formalism. At the core of this chapter, the entangled history of Li’s published algorithms for testing primality and their characteristics in relation to Li’s and his readers’ sources, both ancient and contemporary, Western and Chinese, are discussed. This necessitates an excursion into the number-theoretical books in Euclid’s Elements and their Chinese translation. The chapter ends with a discussion of the related nineteenth- and twentieth-century historiographies of the so-called “Chinese Theorem” in the context of modern, universal mathematics and Fermat’s Little Theorem. In addition, a first full translation of Li Shanlan’s primality test algorithms with numeric examples is given in Appendix B, where symbolic transcriptions of Li’s entirely rhetoric text are added to help understand the mathematical operations involved.
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Notes
- 1.
Translated from Li (1872a) p. 13A.
- 2.
See Li and Wylie (1865).
- 3.
Dated 10 May 1869, quoted from Wylie (1869).
- 4.
The statement is even valid more generally, that is: if N is a prime number, then for every integer a, the number a N − a is divisible by N.
- 5.
- 6.
See Engelfriet (1998).
- 7.
See Xu (2005) p. 16–24.
- 8.
Text according to Billingsley (1570) f186r. Adaptation to modern English spelling is mine, the same holds for all following citations from this book.
- 9.
Chinese text quoted from Li and Wylie (1865) scroll 7, p. 2A, emphasis in red is mine.
- 10.
On the classification of numbers in Greek mathematics, see Heath (1981) vol. 1, p. 70–74, here p. 73–74:
Euclid then, as well as Aristotle, includes 2 among prime numbers. Theon of Smyrna says that even numbers are not measured by the unit alone, except 2, which therefore is odd-like without being prime. The Neo-Pythagoreans, Nicomachus and lamblichus, not only exclude 2 from prime numbers, but define composite numbers, numbers prime to one another, and numbers composite to one another as excluding all even numbers; they make all these categories subdivisions of odd. Their object is to divide odd into three classes parallel to the three subdivisions of even, namely even-even = 2n, even-odd = 2 (2m + 1) and the quasi-intermediate odd-even = 2n+1 (2m + 1); accordingly they divide odd numbers into (a) the prime and incomposite, which, are Euclid’s primes excluding 2, (b) the secondary and composite, the factors of which must all be not only odd but prime numbers, (c) those which are “secondary and composite in themselves but prime and incomposite to another number,” e.g. 9 and 25, which are both secondary and composite but have no common measure except 1. The inconvenience of the restriction in (b) is obvious, and there is the further objection that (b) and (c) overlap, in fact (b) includes the whole of (c).
- 11.
Billingsley (1570) f185v.
- 12.
- 13.
Billingsley (1570) f184r.
- 14.
Chinese text quoted from Li and Wylie (1865) scroll 7, p. 1A.
- 15.
The standard account of the proof of IX.20 is as follows: Assuming that there exists a greatest prime N, since N! + 1 is not divisible by any of the numbers 2 to N, its smallest factor above unity must be a prime greater than N. This contradicts the hypothesis that N is the greatest prime. This is true but it is not what IX.20 proves; it cannot even be reduced to this.
- 16.
Clavius had already formulated this in a scholium as a suggestion to rewrite the proposition as a (construction) problem. See Clavius (1591) p. 77 shown in Fig. 3.6:
Poterat idem hoc theorema instar problematis hoc modo proponi. Primis numeris quotcunque propositis, invenire alium primum numerum ab illis diversum.
- 17.
See Billingsley (1570) f200r:
Proposition (VII.38)
Three numbers being given, to find out the least number which they measure.
- 18.
See Billingsley (1570) f196r:
Proposition (VII.24)
The least numbers that have one and the same proportion with them are prime the one to the other.
It seems that VII.24 is the wrong reference here. It should read “(by the 33. of the seventh)” instead. See Billingsley (1570) f198v:
Proposition (VII.33)
Every composed number is measured by some prime number.
- 19.
Billingsley (1570) f232v.
- 20.
Quoted from Li and Wylie (1865) scrolll 9, p. 15A.
- 21.
Billingsley (1570) f232v.
- 22.
Translated by Fabio Acerbi (personal communication) according to the Greek text in Heiberg’s edition, vol. V, p. 407. The scholium belongs in the oldest layer of annotations to the Elements, since it can be read in MS. Dorvillianus 301, fol. 171r, dated 888 AD (Bodleian Library, Oxford) (see Fig. 3.7) and in MS. Vaticanus Graecus 190, fol. 132r (Biblioteca Apostolica Vaticana).
- 23.
See for example Bréard (1999) p. 194–212.
- 24.
See also Chap. 5 on his work in combinatorics.
- 25.
See Shen (1986).
- 26.
- 27.
The “auxiliary number” is implicitly chosen as coprime to the number N that is to be tested. Since one immediately sees that a number N is not prime when it is even, 2 and an odd number N are always coprime.
- 28.
N = 341 is actually the smallest possible pseudoprime. See page 243.
- 29.
- 30.
The Sunzi suanjing has been transmitted as part of the Tang dynasty collection of Ten Books of Mathematical Classics (Suanjing shi shu 算經十書), in which the Nine Chapters were equally included. For a translation and historiographic discussion of the problem on an unknown number of things and its transmission in versified form, see Bréard (2014) p. 170–173.
- 31.
On the meaning of this technical term in Qin Jiushao’s Dayan procedure to solve indeterminate problems, see Libbrecht (1973) p. 329n15.
- 32.
Translated from Hua (1893b) p. 1A–1B.
- 33.
Chin. yanmu here the product of 2, 3, 5 and 7.
- 34.
Translated from Hua (1893a) p. 31A–31B.
- 35.
Described by Nicomachus in his Introductio Arithmetica, see Nicomachus <Gerasenus> (1866) bk. I, chap. XIII.2-8, p. 29–33 .
- 36.
Idem.
- 37.
Hua (1893a) p. 5A–5B.
- 38.
Li Renshu was the style name of Li Shanlan.
- 39.
- 40.
- 41.
- 42.
See Chap. 5, p. 120.
- 43.
See Volume VI - Issue 214 (1871-06-09) p. 429–430. See also Han and Siu (2008) p. 947–948. On John Fryer’s and Hua Hengfang’s collaboration to translate Western scientific works into Chinese see p. 79.
- 44.
Fryer (1871a) p. 163.
- 45.
For von Gumpach’s date of death, see Hart (1975) vol. 1, p. 203, Letter no. 135.
- 46.
- 47.
See von Gumpach (1870a) and von Gumpach (1869) where he points out p. 153 that Li’s criterion is equivalent to testing if for a given integer number n, the quotient (2n − 2)∕n “is found to be an integer.” Two other responses to Wylie (1869) stem from Mr. W. McGregor of Amoy McGregor (1869) and R. A. Jamieson of Hankow Jamieson (1869), professor of Natural Philosophy and Mathematics in “The University of Peking” at Shanghai and not, as Han and Siu (2008) p. 946 believe, a British journalist. The final note in this series by von Gumpach is von Gumpach (1870b).
- 48.
In a letter to Bernard Frénicle de Bessy (ca. 1605–1675), Pierre Fermat (1607–1665) conjectured the following (Tannery and Henry 1894) p. 209:
Tout nombre premier mesure infailliblement une des puissances − 1 de quelque progression que ce soit, et l’exposant de la dite puissance est sous-multiple du nombre premier donné − 1; et, après qu’on a trouvé la première puissance qui satisfait àla question, toutes celles dont les exposants sont multiples de l’exposant de la première satisfont tout de même à la question. […]
Et cette proposition est généralement vraie en toutes progression et en tous nombres premiers; de quoi je vous envoierois la démonstration, si je n’appréhendois d’être trop long. (Letter from Fermat to Frénicle de Bessy, dated 18 October, 1640)
- 49.
Jamieson (1869) p. 179.
- 50.
von Gumpach (1870b) p. 39.
- 51.
This turned into the famous “von Gumpach case,” a cause célèbre because it was not clear whether the British Supreme Court or the Chinese Government had jurisdiction to deal with it. See Hart (1975) vol. 1, Introduction p. 14 and the many legal notes in the North China Herald between March 29 and May 5, 1870.
- 52.
von Gumpach (1869) p. 153.
- 53.
McGregor (1869) p. 167.
- 54.
See Fryer (1871a) p. 164.
- 55.
Idem.
- 56.
This passage shows that Hua had not understood that Li’s rule is wrong, even for u = 2.
- 57.
Fryer (1871a) p. 164.
- 58.
See page 66.
- 59.
- 60.
That this was not evident to a Chinese mathematician is shown in Hua (1893a) p. 13A, where Hua sets out to deduce that (in modern terms) for k = 1, …, p − 1 the binomial coefficients \(\binom {p}{k}\) are all divisible by p.
- 61.
Fryer (1871a) p. 164.
- 62.
Peano (1901) p. 96–97:
Les chinois ont connu cette P pour b = 2, dès le temps de Confucius. a. —550–477; cfr. Jeans Mm. a.1898 t.27 p. 171.
- 63.
Dickson (2002) vol. 1, p. 59:
The Chinese seem to have known as early as 500 B. C. that 2p − 2 is divisible by the prime p. This fact was rediscovered by P. de Fermat while investigating perfect numbers. Shortly afterwards, Fermat stated that he had a proof of the more general fact now known as Fermat’s theorem: If p is any prime and x is any integer not divisible by p, then x p−1 − 1 is divisible by p.
- 64.
See Needham (1959) p. 54, n.d.
- 65.
See Ribenboim (2004) p. 88–89.
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Bréard, A. (2019). Filling Euclid’s Gaps. In: Nine Chapters on Mathematical Modernity. Transcultural Research – Heidelberg Studies on Asia and Europe in a Global Context. Springer, Cham. https://doi.org/10.1007/978-3-319-93695-6_3
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