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On the youthful writings of Louis J. Mordell on the Diophantine equation \(y^2-k=x^3\)

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Abstract

This article examines the research of Louis J. Mordell on the Diophantine equation \(y^2-k=x^3\) as it appeared in one of his first papers, published in 1914. After presenting a number of elements relating to Mordell’s mathematical youth and his (problematic) writing, we analyze the 1914 paper by following the three approaches he developed therein, respectively, based on the quadratic reciprocity law, on ideal numbers, and on binary cubic forms. This analysis allows us to describe many of the difficulties in reading and understanding Mordell’s proofs, difficulties which we make explicit and comment on in depth.

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Notes

  1. The information of this paragraph comes from the obituary of Mordell written by his former student John Cassels (1973). See also Davenport (1964), Cassels (1986).

  2. About the Sylvester Medal, see Grattan-Guinness (1993), and particularly p. 108 for the quote on Mordell.

  3. In modern terms, the finite basis theorem (sometimes also called “Mordell’s theorem”) states that the group of rational points on an elliptic curve defined over the field of rational numbers is finitely generated. The Mordell–Weil theorem generalizes the latter to the case of an Abelian variety defined over a number field. The Mordell conjecture, now Falting’s theorem, states that the group of rational points on a curve of genus greater than 1 and defined over the rationals is finite.

  4. See for instance Schappacher and Schoof (1996), Roquette (1998), Houzel (2004). Note that Schappacher (1991) devoted a few lines to Mordell’s theorem and explained that this theorem was originally not expressed in group-theoretic terms. Mordell also appears in a few papers which are not dedicated to the history of number theory: see for example Fletcher (1986) about the reaction of Britain to the explusion of mathematicians of the Nazi Germany, Barrow-Green (1999) about the history of the Smith Prize, or Rice and Wilson (2003) about the development of analysis in Britain in the twentieth century. Finally, Mordell’s time in London (1913–1920) has been lately described in Fairbairn (2017).

  5. The lecture has been published two years later under the title A Chapter in the Theory of Numbers (Mordell 1947a). That Mordell came back on the equation \(y^2-k=x^3\) again and again has also been remarked in Cassels (1973, 496) for instance.

  6. This paper is the second item on the list of Mordell’s publications (Cassels 1973, 511–520). As will be seen, the first item (Mordell 1913) is a short note which is actually an outgrowth of the research that led to the 1914 paper.

  7. The Smith’s Prize competition was intended for postgraduates students in Cambridge and consisted in submitting an essay supposed to display the ability of the candidate to produce an original work. At the beginning of the twentieth century, the PhD did not exist in Great Britain and, as June Barrow-Green explains, “the Smith’s Prize competition provides one of the few explicit sources of information about Cambridge postgraduate research in mathematics.” (Barrow-Green 1999, 304).

  8. As of today, the question of determining the parameters k for which the equation has a finite number of rational solutions is still open; it is linked to the Birch and Swinnerton-Dyer conjecture.

  9. The biographical elements of this subsection come from the autobiographical writings (Mordell 1959, 1971). Note that these writings have served to a great extent as a base for later papers on Mordell’s life, such as Davenport (1964), Cassels (1973), Cassels (1974), Fairbairn (2017).

  10. Mordell did not present Ficklin’s book with the same level of precision.

  11. We recall that the Tripos is a competitive exam that was taken by second-year or third-year students of colleges of Cambridge, which thus marked the end of the undergraduate studies. It led to a ranking of students, of which the first ones were called the Wranglers. The ranking was abandoned in 1909, after a reform of the exam. See Wilson (1982), Warwick (2003), Crilly (2011).

  12. See Rice and Wilson (2003). Hardy was a lecturer in Cambridge in these years, but not at St John’s: he entered Trinity College as a student in 1896 and was a lecturer there between 1906 and 1919 (Titchmarsch 1949, 448).

  13. Catherine Goldstein (1999, 396) remarked that between 1870 and 1914, a great part of the English number-theoretic production consisted in elementary papers published in Educational Times, which may explain Mordell’s point of view if the latter aimed at number theory as published in research journals. However, because of the rather small number of historical investigations about number theory in Great Britain at the end of the nineteenth century, it is difficult for us to assess more precisely Mordell’s statements.

  14. Curiously, Mordell did not mention the name of James Whitbread Lee Glaisher, who was still active in number theory at the beginning of the twentieth century. Further, remark that Hardy and John Edensor Littlewood were mostly concerned with questions of analysis at that time.

  15. These ten references about the algebraic theory of forms include a 1903 book written with Alfred Young, entitled The Algebra of Invariants, “which first presented the work of Clebsch and Gordan to British mathematicians.” (Todd 1958, 94). Note that the great number of geometric publications in comparison with the others echoes the usual presentation of Grace as a geometer, (Todd 1958, 94; Barrow-Green and Gray 2006, p. 328 sqq.).

  16. Nevertheless, we will see and discuss the fact that Mordell used “ideal numbers” whereas Mathews (among others) dealt with “ideals.”

  17. Let us add that Henry John Stephen Smith’s Report on the Theory of Numbers is among the recurring references cited by Mordell in his other papers published in the 1910s. Just as with Mathews’ book, Mordell thus could have known this Report when he was studying in Cambridge.

  18. On the contrary, all the theorems and the proofs in the two papers of Hardy published in this volume are clearly delimited, by the use of italics and of labels “Theorem” and “Proof.” More generally, we are not suggesting that the issues about Mordell’s writing that we will describe throughout our paper are either exclusively proper to him or completely widespread in the mathematical literature of the beginning of the twentieth century. Our aim here is to focus on Mordell’s way of writing and its consequences in the context of the paper on \(y^2-k=x^3\), and not to evaluate to what extent such issues could be common at the beginning of the twentieth century, which would lead us far beyond the limits of our paper.

  19. Letter of F. P. White (Honorary Secretary of the London Mathematical Society) to Mordell, dated June 18, 1929, and reporting the comments of an anonymous referee. The letter is kept in Mordell’s papers in the archives of St John’s College (box 2, folder 15).

  20. The letter containing this response is addressed to the Council of the London Mathematical Society. It is kept in Mordell’s papers in the archives of St John’s College (box 6). Note that these archives contain almost no material from the 1910s.

  21. Similar questions about Augustin Louis Cauchy’s research on number theory are raised in Boucard (2013, 363).

  22. In fact, in a 1659 letter to Carcavi (published only at the very end of the nineteenth century), Fermat explained that this problem, together with other “negative” ones, could be solved with the method of infinite descent (Goldstein 1993, 27–28). About Fermat’s works in arithmetic, we refer to the research of C. Goldstein, for instance Goldstein (1995, 2009) . Moreover, in the quotation reported by Mordell, Fermat mentioned that a method of Claude-Gaspard Bachet de Méziriac allowed to find an infinite number of rational solutions to \(y^2+2=x^3\). Nowadays, the equation \(y^2-k=x^3\) is sometimes referred to as the “Bachet equation” or the “Mordell-Bachet equation.”

  23. About this journal, see Rollet and Nabonnand (2013). Jenny Boucard pointed out to us that the subject of cubic and biquadratic Diophantine equations was quite intensively studied in Nouvelles annales between the end of the 1860s and the middle of the 1880s, especially by the people cited by Mordell (Boucard 2019). A doctoral dissertation is currently being prepared on this very subject by Alain Larroche. As for the papers of Pépin cited by Mordell were published in Liouville’s Journal de mathématiques pures et appliquées and in Annales de la Société scientifique de Bruxelles (Pépin 1875, 1882).

  24. About the “questions and solutions” genre in British journals in the eighteenth and nineteenth centuries, see Despeaux (2014).

  25. In other words, this result says that if x and y are coprime, and if q is an odd prime divisor of \(x^2-ky^2\), then \((k/q)=1\). Investigations on such divisors had already been made by Leonhard Euler in the middle of the eighteenth century. These investigations and their link with the reciprocity law are described in Edwards (1983). See also Boucard (2013) for related works of Cauchy on quadratic forms \(x^2+ny^2=4p^\mu \). Moreover, note that Mathews (1892, 52) also dealt with divisors of \(x^2-ky^2\), yet in a different way than Mordell.

  26. This can be seen by writing \(N=\left( x+\frac{ka}{2}\right) ^2+\frac{3}{4}k^2a^2\).

  27. In the course of the preliminary considerations Mordell had proved that if p is an odd prime factor of \(x^2-ky^2\) and if p is prime to ky, then \((k/p)=1\). Here, every prime factor p of N divides \(y^2-klb^2\), so that \((kl/p)=1\) if N is prime to klb (or, equivalently, to \(klb^2\)): this yields \((kl/N)=1\).

  28. Because \(N=x^2+kax+k^2a^2\) is congruent to \(x^2\) modulo k, one has \((N/k)=(x^2/k)=(x/k)^2\). This is equal to 1 since x is prime to k.

  29. Since \(q^2(y_1^2-klb_1^2)=(x-ka)N\), the number \(q^2\) divides N if q is prime to \(x-ka\). Now, write \(N=(x-ka)^2+3kax\): as q divides N, the coprimality of q with 3kax is equivalent to that of q with \(x-ka\).

  30. Just like above, since \((kl/M)=-1\), there exists a prime divisor q of M satisfying \((kl/q)=-1\). Then, as M divides \(Y^2-klB^2\), the preliminary considerations show that q also divides B, whereas B is supposed not to have such a factor. Remark that Mordell’s reasoning can be interpreted as a form of infinite descent, even if Mordell neither made any comment on it, nor even introduced this appellation. As recalled in our introduction, the infinite descent played a crucial role in the proof of the finite basis theorem published in Mordell (1922b). See Goldstein (1993, 40–42).

  31. In this equation, the letters a and b obviously do not have the same meaning as above: the previous a is replaced by \(4a-1\) and the previous b is replaced by 2b. Such shifts in notations are very frequent in Mordell’s text. Moreover, although this was not recalled, the numbers \(4a-1\) and 2b must satisfy the conditions given above, namely that \(4a-1\) and 2b have no common prime factor q such that \((k/q)=-1\) and that \(2b\not \equiv 0\bmod 3\) if \((k/3)=-1\): since Mordell did not rewrite these conditions, one could have thought that the specific, new forms \(4a-1\) and 2b made them tautologically true, but this is not the case.

  32. With the help of a computer we have generated a number of the values given by Mordell’s formulas, and we found other results than those listed by him. For instance, the case of \(y^2=x^3+(4a-1)^3-4b^2\) yields (among others) the insoluble equations \(y^2=x^3-5\) and \(y^2=x^3+23\). All the supplementary values comprised between \(-100\) and 100 that we computed appear in the section dealing with ideal numbers, expect for \(y^2=x^3-33\). Moreover, remark that the last quotation of Mordell is about the equation \(y^2+k=x^3\) and not \(y^2-k=x^3\). Such (exasperating) substitutions between k and \(-k\) repeatedly occur in the paper, for no apparent reason.

  33. Here again, we computed supplementary results, which are all redundant with the values given somewhere else in Mordell’s paper. In particular, Mordell’s formulas for \(l=6\) do give some numerical results between \(-100\) and 100, but they all appear either in the case \(l=2\) or in the next part, dealing with ideal numbers.

  34. The reference given by Mordell is the chapter of Euler’s Elements of Algebra devoted to the more general problem of finding y, z such that \(ay^2+cz^2\) is a perfect power. The mistake of Euler was to think that if \((\sqrt{a}y+\sqrt{-c}z)(\sqrt{a}y-\sqrt{-c}z)\) is a m-th power (of an integer), then the two factors are necessarily themselves m-th powers of complex numbers of the form \(\sqrt{a}p+\sqrt{-c}q\). This mistake is now part and parcel of the history of ideal theory. See Edwards (1980, 323) for instance.

  35. They are both tables displaying solutions of the equations \(t^2-ku^2=\pm 1\) (Legendre 1830, Table X), and \(t^2-ku^2=\pm 4\) (Cayley 1857). We will see later how they intervene in the solution of \(y^2-k=x^3\).

  36. In current notations and terminology, the theory of Kummer deals with algebraic integers of cyclotomic fields \({\mathbf {Q}}(\alpha )\), where \(\alpha \) is a primitive root of unity, while Mordell considers algebraic integers of quadratic fields \({\mathbf {Q}}(\sqrt{k})\), where k is a nonzero integer. On the history of Kummer’s theory, see Edwards (1980). Remark that Edwards (1980, 328) explains that Kummer wanted to generalize his theory to complex numbers of the form \(x+y\sqrt{k}\), but that he never accomplished this generalization.

  37. Moreover, what Mordell does with ideal numbers has nothing in common with Leopold Kronecker’s divisor theory. Kummer, Dedekind, Kronecker, and Hilbert are the principal names that are usually associated with the history of ideal theory. See Edwards (1980), Boniface (2004), as well as Goldstein and Schappacher (2007b, 75–90). Finally, see Schappacher (2005) about the Zahlbericht.

  38. Our decision to use such a modern point of view is also motivated by pedagogical considerations. In particular this choice is not intended to induce any smoothing of the asperities of Mordell’s text that we want to share, a smoothing that any modernized presentation could obviously convey. On the issues raised by the writing of (general) history with a current vocabulary, see Prost (1996, 280–281).

  39. This is the analog of the fundamental theorem of arithmetic in \({\mathbf {Z}}\). An ideal I is said to be prime if, for any \(a,b\in I\), the condition \(ab\in I\) implies that either \(a\in I\) or \(b\in I\). For basic notions of algebraic number theory, see for instance Duverney (2010).

  40. This is a technical result that Mordell would use later. That x is odd can be seen by examining the possible values of \(y^2-k\) and \(x^3\) modulo 4; the coprimality of x and k comes from the hypothesis that k is square-free.

  41. Because the exponent of y in the equation is even, to any solution (x, y) obviously corresponds another solution \((x,-y)\). Mordell counted these solutions as one, and we will use the same convention.

  42. The proof is written in the two lines following the first centered equation in the extract. The coprimality of x and \(2\theta \) comes from the fact that x is prime to 2k: if \(\pi \) divides x and \(2\theta \), then it divides x and \(-2\theta ^2=2k\), so that \(\pi \) is an unit.

  43. We recall that the ideal \(T_1\) is a priori generated by several algebraic integers. By definition, it is principal if it has only one generator, which can be identified to the ideal, up to a unit.

  44. Numbers are said to be associate numbers if they multiplicatively differ from an unit: in \({\mathbf {Z}}[i]\), associate numbers are thus of the form \(a+ib\), \(-(a+ib)\), \(i(a+ib)\), \(-i(a+ib)\). Choosing one privileged number among those four was linked to the issue of formulating a biquadratic reciprocity law. For the usual integers, associate numbers are just opposite numbers, and a canonical way to distinguish one of them is to privilege the positive one.

  45. It is easy to see that the possibilities \(b=\pm 4\) and \(b=\pm 8\) lead to an impossible equality \(8=b[3(2a-b)^2+kb^2]\).

  46. This can be inferred from Mordell (1969, 242), where, in the same proof, U is replaced by \(\pm U\).

  47. Once again, we have no clue of how Mordell ended up considering these specific values of the modulus and these possibilities for \(kf^2\), and we are here reduced to simply check (and complete) his computations.

  48. The adjective “arithmetical,” which was used by Mordell, refers to the fact that all the cubic forms at stake have integral coefficients.

  49. According to the History of the Theory of Numbers of Dickson, the theory of binary cubic forms had been launched by Gotthold Eisenstein with a series of papers published in 1844, which Arndt did not know when he first tackled the same theory in the beginning of the 1850s (Dickson 1923, 255–258). In the 1857 paper cited by Mordell, however, Arndt did refer to the works of Eisenstein and he listed some of their defects. He also explained that his aim was to simplify some of his own past research on cubic forms (Arndt 1857, 309). About Eisenstein, see Schappacher (1998).

  50. See Cayley and Hermite (1857), Mathews (1891) for instance. According to Crilly (2006, 197), the word “Hessian” had been proposed by Sylvester in the beginning of the 1850s. If Mordell’s terminology is the same as that of Cayley and Mathews, his numerical conventions are rather those of Hermite. Whatever the chosen convention and terminology were, the consideration of associated quadratic forms was crucial for those who researched cubics, because it created a bridge between the theory of cubic forms (and, in particular, their classification) and that of quadratic forms, which had been extensively studied since Gauss’s Disquisitiones Arithmeticae (1801). About quadratic forms in the Disquisitiones, see Goldstein and Schappacher (2007a, 8–13). Finally, note that all the elements about quadratic forms that would be used by Mordell appear in Mathews’ Theory of Numbers, and that Pépin (1875) made use of the theory of quadratic forms to tackle the equation \(x^2+ny^2=z^m\).

  51. More generally, invariants and covariants can be defined for forms with non-necessarily integral coefficients, by the invariance up to a power of the determinant of any linear change of variables (with any nonzero determinant). About the history of invariant theory, see Fisher (1966), Crilly (1988), Parshall Hunger (1989).

  52. The cubic covariant T is defined by \(T=(a^2d+2b^3-3abc)x^3+3(b^2c+abd-2ac^2)x^2y+3(2b^2d-bc^2-acd)xy^2+(3bcd-ad^2-2c^3)y^3\). The coefficients of T can be expressed with those of \(\varphi \) and of the Hessian (F, G, H). For instance, the coefficient of \(x^3\) is \(bF-aG\). The cubic covariant T and, a fortiori, the syzygy are not to be found in Arndt’s 1857 paper. According to Meyer (1899, 351), it had been found and proved by Cayley in his Fifth Memoir upon Quantics (Cayley 1858).

  53. If f is the first coefficient of a form \(\varphi \), then \(\varphi (1,0)=f\) is a proper representation of f. Conversely, if \(f=\varphi (u,v)\) is a proper representation of f, it is possible (thanks to Bezout’s identity between u and v) to define a substitution with determinant 1 mapping (u, v) on (1, 0). If \(\varphi \) is transformed into \(\varphi '\) by this substitution, then \(f=\varphi (u,v)=\varphi '(1,0)\), which proves that f is the coefficient of \(x^3\) in \(\varphi '\).

  54. Mordell only suggested that G was supposed to be so chosen that \(fG\equiv -q \bmod F\): this is possible because, as every number x satisfying \(y^2-kf^2=x^3\) is supposed to be prime to f, the numbers f and F are coprime. The form (F, G, H) being supposed to have its determinant equal to k, the last coefficient H must be defined by the relation \(G^2-FH=k\). Thus, there remains to show that F divides \(G^2-k\). But \(f^2(G^2-k)\equiv q^2-kf^2 \equiv 0 \bmod F\), and since \(\gcd (f,F)=1\), this proves that F divides \(G^2-k\).

  55. The same techniques indeed apply to d. That b is an integer comes from the fact that G has been defined through the congruence \(fG\equiv -q \bmod F\). The form (f, b, c, d) being then completely defined, to check that its Hessian is (F, G, H) is a mere computation.

  56. We recall that a quadratic form (a, b, c) is said to be primitive if \(\gcd (a,b,c)=1\), and properly primitive if, additionally, \(\gcd (a,2b,c)=1\). Moreover, the label “class number” usually refers to the number of equivalence classes of (non-necessarily properly primitive) quadratic forms, a number that Gauss had proved to be finite in his Disquisitiones. Since the determinant k is square-free, the number of classes of properly primitive forms is either equal to the number of classes of forms, or to its double, according to whether \(k\equiv 2,3\bmod 4\) or \(k\equiv 1\bmod 4\). Gauss had defined a law of composition for quadratic forms which, in turn, defines a law of composition of classes of quadratic forms: in modern parlance, this makes the set of equivalence classes a group, whose order is the class number.

  57. For instance, \(x^2+y^2\) and \(x^2+2xy+2y^2\) are properly equivalent by the transformation \((x,y)\mapsto (x+y,y)\). This implies that \(5=1^2+2^2\) and \(5=(-1)^2+2\cdot (-1)\cdot 2+2\cdot 2^2\) are counted as one representation of the number 5. Curiously, the number of representations of an integer by forms of given determinant does not seem to have been defined in the references on forms that we have read.

  58. Remark that the syzygy implies that \(T(\xi ,\eta )\) is even for any \(\xi \), \(\eta \). Moreover, the couples \((\xi ,\eta )\) such that \(\varphi (\xi ,\eta )=-f\) give solutions to the equation too. But it can easily be seen that the solution \((H(\xi ,\eta ),T(\xi ,\eta )/2)\) associated with such a couple is equal to \((H(-\xi ,-\eta ),-T(-\xi ,-\eta )/2)\), which is one of the solution associated with the representation \(f=\varphi (-\xi ,-\eta )\). The uncertainty of the sign of the second coordinate of the solutions of \(y^2-kf^2=x^3\) comes from the fact that the exponent of y in the equation is even. As already remarked, Mordell used to talk about unique solutions, thus aiming at solutions (x, y) with \(y>0\). However, it is quite uneasy, in our reconstruction, to introduce such solutions, for nothing is known about the sign of \(T(\xi ,\eta )\) in general. That being said, taking into account both of the sibling solutions (x, y) and \((x,-y)\) does not change anything about the possible finiteness of the set of the solutions.

  59. Mordell’s ignorance of Thue’s paper has been later integrated to the usual story of the equation \(y^2-k=x^3\), by Mordell himself and by others: see for instance Mordell (1971, 957) or Cassels (1986, 35). About Thue’s research, especially on his famous theorem on Diophantine approximation and its consequences, see Goldstein (2015). According to Dickson (1920, 538), Edmund Landau also used Thue’s theorem in a article published in 1920 to prove that \(y^2+2=x^3\) has a finite number of solutions.

  60. To prove that the Hessian is properly primitive is a mere computation: the primitiveness comes from the hypothesis that k is square-free, and the properly primitiveness can be proved with the syzygy and the assumption that every x satisfying \(y^2-kf^2=x^3\) is prime to 2kf. The result stating that Hessians of cubic forms are of order 1 or 3 can be found (for \(k<0\)) in Cayley and Hermite (1857, 85). Arndt’s paper does not contain it. Instead of talking about the order of a (class of) form(s), mathematicians of the time would say that Hessians of cubic forms produce the principal form \((1,0,-k)\) (which is the neutral element in the class group) by triplication.

  61. Gauss had proved that classes of quadratic forms of given determinant k can be grouped into genera, according to certain criteria. In modern parlance, if the genus \(G_0\) containing the principal class is cyclic, the determinant k is said to be regular; if not, it is irregular, and the quotient of the cardinality of \(G_0\) by the maximal order of an element of \(G_0\) is called the exponent (or the index, for Mordell) of irregularity of k. Let us additionally note that Mordell seemed to implicitly define the index of irregularity to be equal to 1 in the case of a regular determinant.

  62. We will see in our final section that this specific cubic form \((1,0,-p,2q)\) would be used by Alan Baker in subsequent works on the equation \(y^2-k=x^3\).

  63. The conditions such as \(kf^2\equiv 4\bmod 9\) are reminiscent of those obtained in the part on ideal numbers, but Mordell did not comment on this.

  64. The only reference whose mathematical objective does not fit in this trichotomy is the paper (Hall 1953), in which Marshall Hall aimed at finding specific conditions on k giving insoluble equations.

  65. See for instance Hemer (1954, 67) or Lal et al. (1966, 322–323).

  66. See Baker (1968), 194–196. More precisely, Baker first established a new proof of Thue’s theorem yielding a boundary for the solutions of equations \(f(x,y)=m\), where f is an irreducible binary form, and combined the techniques of Mordell (1914b) with the theory of reduction of cubic forms as grounded by Hermite and modified later by Davenport. As C. Goldstein pointed out to us, these works are part of those for which he was awarded the Fields Medal in 1970.

  67. Yet another perspective would be given by looking at works dealing with the rational solutions of the equation. In the works bearing on this problem and which are listed in Mordell (1969), the 1914 paper appears neither explicitly, nor implicitly; in particular, the theory of cubic forms as it appeared in this paper is completely absent. The basic tool of the works on the rational problem seems to be the use of a factorization in an algebraic number field, but we do not think that this is to be specifically linked with the 1914 approach of Mordell.

  68. As we said earlier, this appreciation may just reflect that Mordell considered the knowledge on cubic forms that was not in Arndt’s paper as elementary or common knowledge, and other readers could agree with Mordell that “all [that is needed] is contained” in the article of Arndt.

  69. The listed elements may be too common at the beginning of the twentieth century to serve as a proof that Mathew’s book was a source for Mordell, but we recall that the latter cited this book in other papers published in the 1910s, and that he retrospectively listed Mathews as one of the rare number theorist in Great Britain at the time.

  70. Situating a text into (a) network(s) of texts has been the object of some recent research in the history of mathematics. See for instance Goldstein and Schappacher (2007b), Brechenmacher (2007).

  71. On the question of the different interpretations of a text or a theorem by different readers, see Goldstein (1995), Ehrhardt (2012).

  72. On the effect of the construction of various contexts of reading, see Ritter (2004).

References

  • Arndt, F. 1857. Zur Theorie der binären kubischen Formen. Journal für die reine und angewandte Mathematik 53: 309–321.

    Article  MathSciNet  Google Scholar 

  • Baker, A. 1968. The Diophantine Equation \(y^2=x^3+k\). Philosophical Transactions of the Royal Society of London Series A, Mathematical and Physical Sciences 263: 193–208.

    Article  MathSciNet  Google Scholar 

  • Barrow-Green, J. 1999. “A Corrective to the Spirit of too Exclusively Pure Mathematics”: Robert Smith (1689–1768) and his Prizes at Cambridge University. Annals of Science 56 (3): 271–316.

  • Barrow-Green, J. 2014. Cambridge Mathematicians’ Responses to the First World War. In The War of Guns and Mathematics, ed. D. Aubin, and C. Goldstein, 59–124. Mathematical Practices and Communities in France and its Western Allies around World War I. Providence: American Mathematical Society.

  • Barrow-Green, J., and J. Gray. 2006. Geometry at Cambridge, 1863–1940. Historia Mathematica 33: 315–356.

    Article  MathSciNet  MATH  Google Scholar 

  • Boniface, J. 2004. Hilbert et la notion d’existence en mathématiques. Paris: Vrin.

    MATH  Google Scholar 

  • Boucard, J. 2013. Cyclotomie et formes quadratiques dans l’œuvre arithmétique d’Augustin-Louis Cauchy (1829–1840). Archive for History of Exact Sciences 67 (4): 349–414.

    Article  MathSciNet  MATH  Google Scholar 

  • Boucard, J. 2019. A Case Study About Mathematical Journals for Teachers and Students: Number Theory in the Nouvelles annales de mathématiques (1842–1927). Revue d’histoire des mathématiques (to appear).

  • Brechenmacher, F. 2007. L’identité algébrique d’une pratique portée par la discussion sur l’équation à l’aide de laquelle on détermine les inégalités séculaires des planètes (1766–1874). Sciences et techniques en perspective 1: 5–85.

    Google Scholar 

  • Cassels, J.W.S. 1973. Louis Joel Mordell 1888–1972. Biographical Memoirs of Fellows of the Royal Society 19: 493–520.

    Article  Google Scholar 

  • Cassels, J.W.S. 1974. L. J. Mordell. Bulletin of the London Mathematical Society 6: 69–96.

    Article  MathSciNet  MATH  Google Scholar 

  • Cassels, J.W.S. 1986. Mordell’s Finite Basis Theorem Revisited. Mathematical Proceedings of the Cambridge Philosophical Society 100: 31–41.

    Article  MathSciNet  MATH  Google Scholar 

  • Cayley, A. 1857. Note sur l’équation \(x^2-Dy^2=\pm 4\), \(D \equiv 5 ~(\text{ mod } \; 8)\). Journal für die reine und angewandte Mathematik 53: 369–371.

    Article  Google Scholar 

  • Cayley, A. 1858. A Fifth Memoir upon Quantics. Philosophical Transactions of the Royal Society of London 148: 429–460.

    Article  Google Scholar 

  • Cayley, A., and C. Hermite. 1857. On Cubic Forms. The Quarterly Journal of Pure and Applied Mathematics 1: 85–91.

    Google Scholar 

  • Crilly, T. 1988. The Decline of Cayley’s Invariant Theory (1863–1895). Historia Mathematica 15 (4): 332–347.

    Article  MathSciNet  MATH  Google Scholar 

  • Crilly, T. 2006. Arthur Cayley: Mathematician Laureate of the Victorian Age. Baltimore: The John Hopkins University Press.

    MATH  Google Scholar 

  • Crilly, T. 2011. Cambridge. The Rise and Fall of the Mathematical Tripos. In Mathematics in Victorian Britain, ed. R. Flood, A.C. Rice, and R.J. Wilson, 17–32. Oxford: Oxford University Press.

  • Cunningham, A. 1905. Solution to Question 15697. Find all the Integral Solutions, if Possible, of the Equation \(x^2-17=y^3\). Mathematical Questions and Solutions from “The Educational Times” 8: 53–54.

  • Cunningham, A. 1908. Solution to Question 16408. Mathematical Questions and Solutions from “The Educational Times” 14: 106–108.

  • Davenport, H. 1964. L. J. Mordell. Acta Arithmetica 9: 3–12.

    Article  MathSciNet  MATH  Google Scholar 

  • de Jonquières, E. 1878a. Au sujet des cas d’impossibilité d’une solution en nombres entiers de l’équation \(x^3\pm a=y^2\). Nouvelles annales de mathématiques 17: 514–515.

    Google Scholar 

  • de Jonquières, E. 1878b. Détermination de certains cas généraux où l’équation \(x^3\pm a=y^2\) n’admet pas de solution en nombres entiers. Nouvelles annales de mathématiques 17: 374–380.

    Google Scholar 

  • Despeaux, S.E. 2014. Mathematical Questions: A Convergence of Mathematical Practices in British Journals of the Eighteenth and Nineteenth Centuries. Revue d’histoire des mathématiques 20: 5–71.

    MathSciNet  MATH  Google Scholar 

  • Dickson, L.E. 1920. History of the Theory of Numbers: Diophantine Analysis, vol. 2. Washington: The Carnegie Institution of Washington.

    Google Scholar 

  • Dickson, L.E. 1923. History of the Theory of Numbers: Quadratic and Higher Forms, vol. 3. Washington: The Carnegie Institution of Washington.

    MATH  Google Scholar 

  • Duverney, D. 2010. Number Theory. An Elementary Introduction Through Diophantine Problems. Singapore: World Scientific.

    Book  MATH  Google Scholar 

  • Edwards, H.M. 1980. The Genesis of Ideal Theory. Archive for History of Exact Sciences 23: 322–378.

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards, H.M. 1983. Euler and Quadratic Reciprocity. Mathematics Magazine 56 (5): 285–291.

    Article  MathSciNet  MATH  Google Scholar 

  • Ehrhardt, C. 2012. Itinéraires d’un texte mathématique : les réélaborations des écrits d’Évariste Galois au XIX e siècle. Paris: Hermann.

    Google Scholar 

  • Euler, L. 1822. Elements of Algebra, 3rd ed. London: Longman, Rees, Orme, and Co.

    Google Scholar 

  • Fairbairn, B. 2017. Louis Joel Mordell’s Time in London. BSHM Bulletin: Journal of the British Society for the History of Mathematics 32 (2): 160–169.

    Article  MathSciNet  MATH  Google Scholar 

  • Ficklin, J. 1875. The Complete Algebra. New York: Ivison, Blakeman, Taylor & Co.

    MATH  Google Scholar 

  • Fisher, C.S. 1966. The Death of a Mathematical Theory: A Study in the Sociology of Knowledge. Archive for History of Exact Sciences 3 (2): 137–159.

    Article  MathSciNet  MATH  Google Scholar 

  • Fletcher, C.R. 1986. Refugee Mathematicians: A German Crisis and a British Response, 1933–1936. Historia Mathematica 13: 13–27.

    Article  MathSciNet  MATH  Google Scholar 

  • Gauss, C.F. 1876. Carl Friedrich Gauss Werke, vol. 2. Göttingen: Königlichen Gesellschaft der Wissenschaften.

    MATH  Google Scholar 

  • Gauthier, S. 2007. La géométrie des nombres comme discipline (1890-1945). Ph.D. Thesis, Université Pierre et Marie Curie.

  • Gauthier, S. 2009. La géométrie dans la géométrie des nombres : histoire de discipline ou histoire de pratiques à partir des exemples de Minkowski, Mordell et Davenport. Revue d’histoire des mathématiques 15: 183–230.

    MATH  Google Scholar 

  • Gerono, C. 1877. Sur l’impossibilité de résoudre en nombres entiers l’équation \(x^3=y^2+17\). Nouvelles annales de mathématiques 16: 325–326.

    MATH  Google Scholar 

  • Goldstein, C. 1993. Descente infinie et analyse diophantienne : programmes de travail et mise en œuvre chez Fermat, Levi, Mordell et Weil. Cahiers du Séminaire d’histoire et de philosophie des mathématiques 3: 25–49.

    MATH  Google Scholar 

  • Goldstein, C. 1995. Un théorème de Fermat et ses lecteurs. Saint-Denis: Presses Universitaires de Vincennes.

    MATH  Google Scholar 

  • Goldstein, C. 1999. Sur la question des méthodes quantitatives en histoire des mathématiques: le cas de la théorie des nombres en France (1870–1914). Acta historiae rerum naturalium necnon technicarum 3: 187–214.

    Google Scholar 

  • Goldstein, C. 2009. L’arithmétique de Fermat dans le contexte de la correspondance de Mersenne: une approche micro-sociale. Annales de la faculté des sciences de Toulouse 18: 25–57.

    Article  MATH  Google Scholar 

  • Goldstein, C. 2015. Axel Thue in Context. Journal de théorie des nombres de Bordeaux 27: 309–337.

    Article  MathSciNet  MATH  Google Scholar 

  • Goldstein, C., and N. Schappacher. 2007a. A Book in Search of a Discipline. In The Shaping of Arithmetic After C. F. Gauss’s Disquisitiones Arithmeticae, ed. C. Goldstein, N. Schappacher, and J. Schwermer, 3–65. Berlin: Springer.

  • Goldstein, C., and N. Schappacher. 2007b. Several Disciplines and a Book. In The Shaping of Arithmetic After C. F. Gauss’s Disquisitiones Arithmeticae, ed. C. Goldstein, N. Schappacher, and J. Schwermer, 67–103. Berlin: Springer.

  • Grattan-Guinness, I. 1993. The Sylvester Medal: Origins, and Recipients 1901–1949. Notes and Records of the Royal Society of London 47 (1): 105–108.

    Article  MathSciNet  MATH  Google Scholar 

  • Hackley, C.W. 1849. A Treatise on Algebra, 3rd ed. New York: Harper & Brothers.

    Google Scholar 

  • Hall, M. 1953. Some Equations \(y^2=x^3-k\) Without Integer Solutions. Journal of the London Mathematical Society 28: 379–383.

    Article  MathSciNet  MATH  Google Scholar 

  • Hemer, O. 1954. Notes on the Diophantine Equation \(y^2-k=x^3\). Arkiv för Matematik 3: 67–77.

    Article  MathSciNet  MATH  Google Scholar 

  • Houzel, C. 2004. Poincaré et l’analyse diophantienne. In De Zénon d’Elée à Poincaré : recueil d’études en hommage à Roshdi Rashed, ed. R. Morelon, and A. Hasnaoui, 221–236. Leuven: Peeters Leuven.

    Google Scholar 

  • Lal, M., M.F. Jones, and W.J. Blundon. 1966. Numerical Solutions of the Diophantine Equation \(y^3-x^2=k\). Mathematics of Computation 20: 322–325.

    MathSciNet  MATH  Google Scholar 

  • Lebesgue, V.A. 1869. Note sur quelques équations indéterminées. Nouvelles annales de mathématiques 8: 452–456.

    Google Scholar 

  • Legendre, A.M. 1830. Théorie des nombres, vol. 1, 3rd ed. Paris: Firmin-Didot.

  • Lejeune-Dirichlet, J.P.G. 1828. Mémoire sur l’impossibilité de quelques équations indéterminées du cinquième degré. Journal für die reine und angewandte Mathematik 3: 354–375.

    Article  MathSciNet  Google Scholar 

  • Ljunggren, W. 1963. On the Diophantine Equation \(y^2-k=x^3\). Acta Arithmetica 8: 451–463.

    Article  MathSciNet  MATH  Google Scholar 

  • Mathews, G.B. 1891. Irregular Determinants and Subtriplicate Forms. The Messenger of Mathematics 20: 70–74.

    Google Scholar 

  • Mathews, G.B. 1892. Theory of Numbers. Cambridge: Deighton, Bell and Co.

  • Mathews, G.B. 1893a. On the Algebraical Integers Derived from an Irreducible Cubic Equation. Proceedings of the London Mathematical Society 24: 327–336.

    MathSciNet  MATH  Google Scholar 

  • Mathews, G.B. 1893b. On the Complex Integers Connected with the Equation \(\theta ^3-2=0\). Proceedings of the London Mathematical Society 24: 319–327.

    MathSciNet  MATH  Google Scholar 

  • Mathews, G.B. 1911. Relations Between Arithmetical Binary Cubic Forms and Their Hessians. Proceedings of the London Mathematical Society 9: 200–204.

    Article  MathSciNet  MATH  Google Scholar 

  • Meyer, W.F. 1899. Invariantentheorie. In: Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol I. B. 2, Teubner, pp. 320–403.

  • Mordell, L.J. 1913. Note on Irregular Determinants. Messenger of Mathematics 42: 124.

    Google Scholar 

  • Mordell, L.J. 1914a. Indeterminate Equations of the Third and Fourth Degrees. The Quarterly Journal of Pure and Applied Mathematics 45: 170–186.

    MATH  Google Scholar 

  • Mordell, L.J. 1914b. The Diophantine Equation \(y^2-k=x^3\). Proceedings of the London Mathematical Society 13: 60–80.

    Article  MathSciNet  Google Scholar 

  • Mordell, L.J. 1915. Theta Functions in the Theory of the Modular Functions and Groups of Linear Substitutions Arising Therefrom. The Quarterly Journal of Pure and Applied Mathematics 46: 97–124.

    MATH  Google Scholar 

  • Mordell, L.J. 1918. The Theory of Numbers. Science Progress 12: 127–131.

    Google Scholar 

  • Mordell, L.J. 1920. A Statement by Fermat. Proceedings of the London Mathematical Society 18: v–vi.

    MATH  Google Scholar 

  • Mordell, L.J. 1922a. Note on the Integer Solutions of the Equation \(Ey^2=Ax^3+Bx^2+Cx+D\). Messenger of Mathematics 51: 169–171.

    Google Scholar 

  • Mordell, L.J. 1922b. On the Rational Solutions of the Indeterminate Equations of the Third and Fourth Degrees. Proceedings of the Cambridge Philosophical Society 21: 179–192.

    MATH  Google Scholar 

  • Mordell, L.J. 1923b. On the Integer Solutions of the Equation \(ey^2=ax^3+bx^2+cx+d\). Proceedings of the London Mathematical Society 21: 415–419.

    Article  MathSciNet  MATH  Google Scholar 

  • Mordell, L.J. 1947a. A Chapter in the Theory of Numbers. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Mordell, L.J. 1947b. On Some Diophantine Equations \(y^2=x^3+k\) with no Rational Solutions. Archiv for Mathematik og Naturvidenskab 49: 143–150.

    MathSciNet  MATH  Google Scholar 

  • Mordell, L.J. 1959. Reflections of a Mathematician. Cambridge: Cambridge University Press.

    MATH  Google Scholar 

  • Mordell, L.J. 1966. The Infinity of Rational Solutions of \(y^2=x^2+k\). Journal of the London Mathematical Society 41: 523–525.

    Article  MathSciNet  MATH  Google Scholar 

  • Mordell, L.J. 1968. On some Diophantine Equations \(y^2=x^3+k\) with no Rational Solutions (II). Abhandlungen aus Zahlentheorie und Analysis. Zur Erinnerung an Edmund Landau, 225–232. Berlin: VEB Deutscher Verlag der Wissenschaften.

  • Mordell, L.J. 1969. Diophantine Equations. London and New York: Academic Press.

    MATH  Google Scholar 

  • Mordell, L.J. 1971. Reminiscences of an Octogenarian Mathematician. The American Mathematical Monthly 78: 952–961.

    Article  MathSciNet  MATH  Google Scholar 

  • Nagell, T. 1930. Einige Gleichungen von der Form \(ay^2+by+c=dx^3\). Avhandlinger utgitt av det Norske Videnskaps-Akademi i Oslo I: Matematisk-naturvidenskapelig Klasse 7: 3–15.

  • Parshall Hunger, K. 1989. Toward a History of Nineteenth-Century Invariant Theory. In The History of Modern Mathematics. Ideas and their Reception, vol. 1, ed. J. McCleary, and D.E. Rowe, 157–206. New York: Academic Press.

  • Pépin, T. 1875. Sur certains nombres complexes compris dans la formule \(a+b\sqrt{-c}\). Journal de Mathématiques pures et appliquées 1: 317–372.

    MATH  Google Scholar 

  • Pépin, T. 1882. Sur le problème de former un carré en ajoutant un cube à un nombre donné. Annales de la société scientifique de Bruxelles 6: 86–100.

    MATH  Google Scholar 

  • Prost, A. 1996. Douze leçons sur l’histoire. Paris: Éditions du Seuil.

    Google Scholar 

  • Réalis, S. 1883. Sur une équation indéterminée. Nouvelles annales de mathématiques 2: 289–297.

    MATH  Google Scholar 

  • Rice, A.C., and R.J. Wilson. 2003. The Rise of British Analysis in the Early 20th Century: The Role of G.H. Hardy and the London Mathematical Society. Historia Mathematica 30: 173–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Ritter, J. 2004. Reading Strasbourg 368: A Thrice-Told Tale. In History of Science, History of Text, ed. K. Chemla, 177–200. Dordrecht: Springer.

    Chapter  Google Scholar 

  • Rollet, L., and P. Nabonnand. 2013. Un journal pour les mathématiques spéciales : les Nouvelles annales de mathématiques (1842–1927). Bulletin de l’Union des Professeurs de Spéciale 86: 5–18.

    Google Scholar 

  • Roquette, P. 1998. Zur Geschichte der Zahlentheorie in den dreißiger Jahren. Mathematische Semesterberichte 45 (1): 1–38.

    Article  MathSciNet  MATH  Google Scholar 

  • Schappacher, N. 1991. Développement de la loi de groupe sur une cubique. In: Goldstein C. (ed) Séminaire de théorie des nombres de Paris 1988/89, Birkhäuser, vol. 91, pp. 159–184.

  • Schappacher, N. 1998. Eisenstein. In Mathematics in Berlin, ed. Begehr H. J. Kramer, H. Koch, N. Schappacher, and E.J. Thiele, 55–60. Basel, Boston, Berlin: Birkhäuser

  • Schappacher, N., 2005. David Hilbert, Report on Algebraic Number Fields (‘Zahlbericht’), 1897. In Landmark Writings in Western Mathematics, 1640–1940, ed. I. Grattan-Guinness, 700–709. Amsterdam: Elsevier.

  • Schappacher, N., and R. Schoof. 1996. Beppo Levi and the Arithmetic of Elliptic Curves. The Mathematical Intelligencer 18 (1): 57–69.

    Article  MathSciNet  MATH  Google Scholar 

  • Smith, H.J.S. 1860. Report on the Theory of Numbers, Part II. Report of the British Association for the Advancement of Science pp. 120–169, Reproduced in The Collected Mathematical Papers, James Whitbread Lee Glaisher (ed.), Oxford: Clarendon Press, pp. 93–162, 1894.

  • Sommer, J. 1907. Vorlesungen über Zahlentheorie: Einfuhrung in die Theorie der algebraischen Zahlkörper. Leipzig, Berlin: Teubner.

    MATH  Google Scholar 

  • Thue, A. 1909. Über Annäherungswerte algebraischer Zahlen. Journal für die reine und angewandte Mathematik 135: 284–305.

    MathSciNet  MATH  Google Scholar 

  • Titchmarsch, E.C. 1949. Godfrey Harold Hardy, 1877–1947. Obituary Notices of Fellows of the Royal Society 6: 446–461.

    MathSciNet  Google Scholar 

  • Todd, J.A. 1958. John Hilton Grace 1873–1958. Biographical Memoirs of Fellows of the Royal Society 4: 92–97.

    Article  Google Scholar 

  • Warwick, A. 2003. Masters of Theory. Cambridge and the Rise of Mathematical Physics. Chicago: The University of Chicago Press.

    Book  MATH  Google Scholar 

  • Wilson, D.B. 1982. Experimentalists among the Mathematicians: Physics in the Cambridge Natural Sciences Tripos, 1851–1900. Historical Studies in the Physical Sciences 12 (2): 325–371.

    Article  Google Scholar 

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Acknowledgements

We would like to thank Jeremy Gray for his valuable suggestions which allowed us improve the paper, as well as for his precious linguistic help. We are also grateful to Catherine Goldstein and Norbert Schappacher for the discussions they had with us about the content of this paper.

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Gauthier, S., Lê, F. On the youthful writings of Louis J. Mordell on the Diophantine equation \(y^2-k=x^3\). Arch. Hist. Exact Sci. 73, 427–468 (2019). https://doi.org/10.1007/s00407-019-00231-1

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