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Historical and Foundational Details on the Method of Infinite Descent: Every Prime Number of the Form 4n + 1 is the Sum of Two Squares

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Abstract

Pierre de Fermat (1601/7–1665) is known as the inventor of modern number theory. He invented–improved many methods useful in this discipline. Fermat often claimed to have proved his most difficult theorems thanks to a method of his own invention: the infinite descent (Fermat 1891–1922, II, pp. 431–436). He wrote of numerous applications of this procedure. Unfortunately, he left only one almost complete demonstration and an outline of another demonstration. The outline concerns the theorem that every prime number of the form 4n + 1 is the sum of two squares. In this paper, we analyse a recent proof of this theorem. It is interesting because: (1) it follows all the elements of which Fermat wrote in his outline; (2) it represents a good introduction to all logical nuances and mathematical variants concerning this method of which Fermat spoke. The assertions by Fermat will also be framed inside their historical context. Therefore, the aims of this paper are related to the history of mathematics and to the logic of proof-methods.

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Notes

  1. de Fermat’s date (hereafter Fermat) of birth seems to be commonly accepted as established in 1601. But, Barner (2001) indicates that such a date is more probably 1607 or 1608. We assume no position on that.

  2. […] tout nombre premier, qui surpasse l'unité d'un multiple de 4, est composé de deux quarrés […]” (Fermat 1891–1922, II, p. 432). This 4n+1 theorem is also called Girard's theorem. See below.

  3. [1891] Œuvres de Fermat, t. I, Œuvres mathématiques diverses – Observations sur Diophante, éd. P. Tannery et C. Henry, Paris, Gauthier-Villars; [1894], Œuvres de Fermat, t. II, Correspondance, éd. P. Tannery et C. Henry, Paris, Gauthier-Villars; [1896], Œuvres de Fermat, t. III, Traductions des écrits latins de Fermat; de l'Inventum novum de J. de Billy; du Commercium epistolicum de Wallis par P. Tannery, éd. P. Tannery et C. Henry, Paris, Gauthier-Villars; [1912], Œuvres de Fermat, t. IV, Compléments par P. Tannery, éd. P. Tannery et C. Henry, Paris, Gauthier-Villars; [1922], Œuvres de Fermat, supp. T. I-IV par M. C. de Waard, éd. P. Tannery et C. Henry, Paris, Gauthier-Villars.

  4. Generally speaking, it depends on the congruence to 0 or to − 1 (mod. 4).

  5. One might also claim that another smaller number exists, which does not have that property. But n is an arbitrary number, so descending infinitely to all n–positive integers, one arrives at n = 1. In other words, the descending sequence starting from p has, so to say, a natural less number, that is the smallest number of the prime 4n + 1, namely 5, assuming n = 1.

  6. The problem of the logical relations between infinite descent and the various forms of mathematical induction (ordinary mathematical induction, Noetherian induction, and so on) will be faced if the sixth section of this paper. The three items we add in the running text need only as a description of the differences between the mathematical application of infinite descent and ordinary induction. For the moment we do not enter the logical questions connected to the relations between the two methods. In addition and generally speaking, it is not necessary to claim a specific case for which the theorem is satisfied; it is only necessary to prove that the basic case (n = 1) contradicts.

  7. One of us (PB) translated Paolini’s work into English (Bussotti 2006, pp. 481–554; for the proofs on the binary quadratic forms, see pp. 481–507, pp. 496–499; on Fermat see pp. 17–184). On Paolini see: Bussotti and Paolini 1997; Bussotti 2000; Bussotti 2008, pp. 63–112.

  8. From a theoretical point of view, it would be sufficient to show that, if the theorem was false, then more than \(m - n - 1\) numbers would exist between m and n, which is absurd.

  9. With regard to the discovery of this letter by Fermat to Huygens, see: Henry 1879 in Fermat 1879, pp. 737–740; Bussotti 2006, p. 5, ft. 4.

  10. “1. Et pour ce que les méthodes ordinaires, qui sont dans les Livres, étoient insuffisantes à démontrer des propositions si difficiles, je trouvai enfin une route tout à fait singulière pour y parvenir. J'appelai cette manière de démontrer la descente infinie ou indéfinie, etc.” je ne m'en servis au commencement que pour démontrer les propositions négatives, comme, par exemple: […] Qu’il y a aucun triangle rectangle en nombres dont l'aire soit un nombre quarré […] (Fermat 1891–1922, II, p. 431; see also pp. 212–217). Our translation.

  11. Fermat did not write the word ordinary. We use it because it expresses epistemologically–synthetically the thought of Fermat that there are four different ways in which his method can be applied.

  12. This proposition is the only one of which Fermat left an almost complete demonstration in his Observations sur Diophante (observation 45; Fermat 1891–1922, I, p. 340. See also Bussotti and Paolini 1997, pp. 36–39 and 55–71; Edwards 1977, chapter 1.6.; Goldstein 1995; Mahoney 1973, 1994, pp. 352–354; Weil 1984, chapter 2, paragraph X.

  13. To write a class of numbers in the form kn + h means, in modern terms, to write such a class in function of the modulus k and the remainder or residue h. Fermat, Euler, Lagrange and Legendre were well aware of this way of writing and of the concepts of modulus and of residue. However, the mathematician who fully developed the whole potential of the notion of congruence based and a modulus and on a residue was Gauss in his Disquisitiones Arithmeticae (Gauss 1801). Gauss gave the formal definition of two congruent numbers in respect to a modulus (first section of the Disquisitiones) and based the entire, magnificent theory expounded in his masterpiece on the concept of congruence between two numbers.

  14. “3. Il y a infinies questions de cette espèce […]” (Fermat 1891–1922, II, p. 432). Our translation.

  15. He repeated these theorems concerning the decomposition of the primes of the forms 4n + 1, 6n + 1, 8n + 1, 8n + 3 in a letter to Digby on June 1658 (Fermat 1891–1922, II, p. 403; see also Bussotti 2006, pp. 177–180).

  16. Fermat dealt with the polygonal numbers theorem on many occasions. The general proposition (every integer is the sum of three triangulars, of four squares, of five pentagonals, of six hexagonals, on so on) was, for example, mentioned in Observations sur Diophante (Fermat 1891–1922, I, p. 305), in a letter to Mersenne in September/October 1636 (Fermat 1891–1922, II, pp. 65–66). In the letter to Pascal on 25 September 1654 (Ivi, pp. 312–313). In the letter to Digby on 19 June 1658 (Ivi, pp. 403–404). For the proofs given by Paolini of the three triangulars and four squares theorem with methods available to Fermat, see Paolini (Bussotti 2006, Appendix, pp. 507–534 and 534–547 respectively). For the explanation of the used methods, see: Bussotti 2006, pp. 109–171. The reference to Pell equation dates to a late phase of Fermat’s “mathematical career”. Beyond the letter to Huygens, Fermat spoke of this equation starting from February 1657, for example in a letter to Frenicle in that month (Fermat, 1891–1922, II, p. 333). A letter to Brouncker dates to the same month (Ivi, p. 335). For the relations between Fermat and the English mathematicians as to the solution of this equation see Bussotti 2006, pp. 77–109. On so called Pell–Fermat equation see: Barbeau 2003; Hofmann 1944; Konen 1901; Selenius 1963; Weil 1977.

  17. John Pell (1611–1685) has nothing to do with the equation. The name was inaccurately attributed by Euler (Euler 1765).

  18. It is well known that Fermat mentioned more than once the impossibility to solve in integers the two equations \(x^{3} + y^{3} = z^{3}\) and \(x^{4} + y^{4} = z^{4}\), but he mentioned the impossibility to solve in integers the general equation \(x^{n} + y^{n} = z^{n}\)—apart from the trivial solutions – only in Observations sur Diophante, question 2 (Fermat 1891–1922 I, p. 291). The proof of the impossibility to solve in integers the equation \(x^{4} + y^{4} = z^{4}\) (to be precise \(x^{4} + y^{4} = z^{2}\)) is included in the theorem that no Pythagorean triangle has the area equal to the square of an integer, while we have no demonstration left by Fermat of the impossibility to solve in integers the equation \(x^{3} + y^{3} = z^{3}\).

  19. See also Goldbach 1747, April 6th and 1749, April 12th; Euler 1752–1753, pp. 3–40, 1754–1755, pp. 3–13.

  20. Euler used two methods to prove these theorems on the binary quadratic form \(x^{2} + Ay^{2}\) (A = 1, 2, 3). We call a first demonstration (see above Sect. 5.2) reduction-descent. It was given for the form \(x^{2} + y^{2}\) (Euler 1752–1753); form \(x^{2} + 2y^{2}\) (Euler 1756-1757); form \(x^{2} + 3y^{2}\) (Euler 1760–1761). We call a second version of the proof as ordinary reduction (see below; Euler 1773).

  21. It is well known that Euler’s proof is based on some assumptions not demonstrated by Euler. Anyway, for the assumed assumptions it is possible to make Euler’s proof more rigorous. Euler used \(Q\left( {\sqrt { - 3} } \right)\). Johann Carl Friedrich Gauß (1777–1855) proved this theorem by means of a reversed induction using \(Z\left( {\sqrt[3]{1}} \right)\). Cfr. Gauss posthumous works (Gauss posthumous, Werke II, pp. 387–391). Weil reworked Euler’s proof without using \(\sqrt { - 3}\) (Weil 1984, chapter 1, paragraph XVI; see also Bussotti 2006, pp. 279–287; Macys 2007). For commentary on Gauss’ proof see Dickson [1919] 1920 [1923], p. 548; Ribenboim 1979, p. 39; Bussotti 2006, pp. 434–437. For reconstructions based on the descent but on principles different from Euler’s see: Paolini (Bussotti 2006, Appendix, pp. 547–554; pp. 171–176) and Piyadasa (Piyadasa 2010; in this proof \(\sqrt[3]{1}\) is used).

  22. As to the works dedicated to the the sums of four squares (Cfr. Euler 1754–1755; see also Pieper 1993; Bussotti 2006, pp. 261–273).

  23. As to Lagrange’s demonstration of the four squares theorem see also Boucard 2014. For a story of the polygonal number theorem from the Greek period to Cauchy, also including Gauss’ proof that every integer is the sum of three triangulars see Bussotti and Scimone 2009. With regard to Lagrange’s Recherches d’arithmétique and the use of the descent, see: Lagrange 1773–1775, pp. 723–737; Bussotti 2006, pp. 362–396; Pisano and Capecchi 2013.

  24. For the theorems proved by descent, see i.e.,: Bussey 1918; Bussotti and Paolini 1997; Bussotti 2000; Bussotti 2006; Cassinet 1980; Conrad (s.d.); Dickson [1919] 1920 [1923] (many references); Genocchi 1855; Genocchi 1883; Hofmann 1960–1962; Lemmermeyer 2003; Piyadasa 2010; Shirali 2003; Tat–Wing 2005; Vacca 1927-1928; Vandiver 1932. For interesting methodological considerations see Brotherston and Simpson 2007; Smith 1992, Wirth 2004, Wirth 2010.

  25. For a research concerning the applications in Fermat, Euler, Lagrange and Gauss see Bussotti (Bussotti 2006). On Lagrange methods see Pisano and Capecchi (2013).

  26. We have slightly modified the definition. Paolini offered two proofs of the 4n + 1 primes theorem: the former (Ivi, pp. 492–495) based on the concept of even continued fraction is inspired by Lucas (Lucas 1891, pp. 250–251). The latter is the one we are expounding. This is easier than the former and can be generalized to other binary quadratic forms, whereas the former is valid only for the primes of the form 4n + 1.

  27. Theorem 2 (Bussotti 2006, pp. 488–490 for the first version; p. 497 for the second one). Theorem 3 (Ivi, pp. 491–492 for the first version; pp. 497–498 for the second one). Theorem 4 (Ivi, pp. 498–499).

  28. The succession of the \(x_{n}\) is given by x-k = \(x_{1}\), x − 2 k = \(x_{2}\),…, x-nk = \(x_{n}\).

  29. So, for example, starting from \(2 \cdot 41 = 9^{2} + 1\), we obtain \(\begin{aligned}2 \cdot 25 = (9 - 2)^{2} + 1 = 7^{2} + 1 \\ 2 \cdot 13 = (7 - 2)^{2} + 1 = (9 - 2 \cdot 2)^{2} + 1 = 5^{2} + 1 \\ 2 \cdot 5 = (5 - 2)^{2} + 1 = (7 - 2 \cdot 2)^{2} + 1 = (9 - 2 \cdot 3)^{2} + 1 = 3^{2} + 1\end{aligned}.\)

  30. We have here summarized a reasoning which is rather refined and which – in the form given by Euler – is not completely satisfying, but the basis of Euler’s argument is correct at all. On the forms \(a^{2} + b^{2}\), \(a^{2} + 2b^{2}\) and \(a^{2} + 3b^{2}\) (Bussotti 2006, pp. 222–226; 238–242; p. 246 respectively).

  31. The features of this section are different from those of the other sections, where we have presented the final results of a research, while this section has to be interpreted as an outline and a proposal for a new research rather than the explanation of final results.

  32. See, for example the admirable demonstrations by Lagrange that all the solutions of Pell equation \(t^{2} - Du^{2} = 1\) are of the form \(t = \frac{{(t_{1} + u_{1} \sqrt D )^{m} + (t_{1} - u_{1} \sqrt D )^{m} }}{2}\); \(u = \frac{{(t_{1} + u_{1} \sqrt D )^{m} + (t_{1} - u_{1} \sqrt D )^{m} }}{{2\sqrt[{}]{D}}}\) (Lagrange 1774, Sects. 72–75; Bussotti 2006, pp. 352–362).

  33. “La proposition fondamentale des triangles rectangles est que tout nombre premier, qui surpasse de l'unité un multiple de 4, est compose de deux quarrés”. (Fermat 1891–1922, II, p. 221; Author’s translation).

References

  • Barbeau, E. (2003). Pell’s equation. New York: Springer.

    Book  Google Scholar 

  • Barner, K. (2001). Das Leben Fermats. DMV Mitteilungen, 3, 12–26.

    Google Scholar 

  • Bellissima, F., & Pagli, P. (1996). Consequentia mirabilis: una regola logica tra matematica e filosofia. Olschki: Firenze.

    Google Scholar 

  • Boucard, J. (2014). Joseph-Louis Lagrange e il teorema dei quattro quadrati. Lettera Matematica Pristem, 88(89), 59–69.

    Google Scholar 

  • Brotherston, J., Simpson, A. (2007). Complete Sequent Calculi for Induction and Infinite Descent. In Proceedings of logic in computer science. 22nd annual IEEE symposium on logic in computer science (LICS 2007) (pp. 51-62). Los Alamos: IIEE.

  • Bussey, W. H. (1918). Fermat’s method of infinite descent. The American Mathematical Monthly, 25(8), 333–337.

    Article  Google Scholar 

  • Bussotti, P. (2000). “Ogni numero primo della forma 4n + 1 è la somma di due quadrati”: storia dei metodi dimostrativi usati per provare un teorema. L’insegnamento della matematica e delle scienze integrate, 23A, 27–63.

    Google Scholar 

  • Bussotti, P. (2006). From Fermat to Gauss. Indefinite descent and methods of reduction in number theory. Augsburg: Rauner.

    Google Scholar 

  • Bussotti, P. (2008). Problems and methods at the origin of the theory of numbers. Napoli: Giannini.

    Google Scholar 

  • Bussotti, P., & Paolini, S. (1997). Pierre de Fermat e la discesa indefinite. Pisa: ETS.

    Google Scholar 

  • Bussotti, P., & Pisano, R. (2014). Newton’s Philosophiae Naturalis Principia Mathematica “Jesuit” Edition: The Tenor of a Huge Work. Accademia Nazionale Lincei Rendiconti Lincei Matematica e Applicazioni, 25(4), 413–444.

    Article  Google Scholar 

  • Bussotti, P., & Scimone, A. (2009). Sulle orme di Fermat. Il teorema dei numeri poligonali e la sua dimostrazione. Lugano: Agorà-Lumieres internationales.

    Google Scholar 

  • Cassinet, R. (1980). Histoire de la descente infinie da Campanus à Hilbert. Cahiers du Séminaire d’Histoire des Mathématiques de Toulouse, 2/B, 1–25.

    Google Scholar 

  • Conrad, K. (s. d). Proofs by descent. Retrieved http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/descent.pdf.

  • de Fermat, P. (1891–1922). Œuvres, four volumes plus Supplément aux tomes I–IV. Paris: Gauthier–Villars.

  • Dickson, L. E. ([1919] 1920 [1923]). History of the theory of numbers (Vol. 3 Vol). Washington: Carnegie Institution of Washington.

  • Edwards, H. M. (1977). Fermat’s last theorem: A genetic introduction to algebraic number theory. New York: Springer.

    Book  Google Scholar 

  • Euler, L. (1732–1738). Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus. Commentarii academiae scientiarum Petropolitanae 6, pp. 103–107. In OO, Ser. I, II, pp. 1–5.

  • Euler, L. (1752–1753). De numeris qui sunt aggregata duorum quadratorum. Novi commentarii academiae scientiarum Petropolitanae, 4, pp. 3–40. In OO, Ser. 1, II, pp. 295–327.

  • Euler, L. (1754–1755). Demonstratio theorematis Fermatiani omnem numerum sive integrum sive fractum esse summam quatuor pauciorumve quadratorum. Novi commentarii academiae scientiarum Petropolitanae 5, pp. 13–58. In OO, Ser. 1, II, pp. 338–372.

  • Euler, L. (1756–1757). Specimen de usu observationum in mathesi pura. Novi commentarii academiae scientiarum Petropolitanae, 6, pp. 185–230. In OO, Ser. 1, II, pp. 459–492.

  • Euler, L. (1760–1761). Supplementum quorundam theorematum arithmeticorum quae in nonnullis demonstrationibus supponuntur. Novi commentarii academiae scientiarum Petropolitanae, 8, pp. 105–128. In OO, Ser. 1, II, pp. 556–575.

  • Euler, L. (1765). De usu novi algorithmi in problemate Pelliano solvendo. Novi commentarii academiae scientiarum Petropolitanae, 11, pp. 28–66. In OO, Ser. 1, III, pp. 73–111.

  • Euler, L. (1770). Algebra. Petersburg. In OO, Ser. 1, I.

  • Euler, L. (1773). Novae demonstrationes circa resolutionem numerorum in quadrata. Nova acta eruditorum, pp. 193–211. In OO, Ser. 1, III, pp. 218–239.

  • de Fermat, P. (1879). Relation des nouvelles découvertes en la science des nombres. Bullettino di bibliografia, XII, 737–740.

    Google Scholar 

  • Gauss, C. F. (1801). Disquisitiones arithmeticae. Leipzig. In Gauss [1870] 1876 (Vol. I).

  • Gauss, C. F. ([1870] 1876). Werke (Vol. I–II). Göttingen-Berlin, Gesellschaft der Naturwissenschaften zu Göttingen.

  • Gauss, C. F. (posthumous). Zur Theorie der complexen Zahlen. Neue Theorie der Zerlegung der Cuben (Nachlass). In Gauss [1870] 1876 (Vol. I, pp. 387–391).

  • Gauthier, Y. (1991). La logique interne. Paris: Vrin.

    Google Scholar 

  • Gauthier, Y. (2002). Internal logic. Foundations of mathematics from Kronecker to Hilbert. Dordrecht: Kluwer.

    Google Scholar 

  • Genocchi, A. (1855). Sopra tre scritti inediti di Leonardo Pisano pubblicati da B. Boncompagni. Annali di scienze matematiche e fisiche VI. Roma, Tipografia delle Belle Arti, pp. 161–185, pp. 218–251, pp. 273–320, pp. 345–362.

  • Genocchi, A. (1883). Intorno a un manoscritto di Pietro Fermat testè pubblicato. Rivista scientifico–industriale delle principali scoperte ed invenzioni fatte nelle scienze e nelle industrie (pp. 148–151) [French translation: Fermat, Oeuvres, supplément aux tomes I–IV, pp. 152–157].

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

    Google Scholar 

  • Hofmann, J. E. (1944). Studien zur Zahlentheorie Fermats (Über die Gleichung x2 = py2 + 1). Berlin: Verlag der Akademie der Wissenschaften.

    Google Scholar 

  • Hofmann, J. E. (1960-62). Über zahlentheoretische Methoden Fermats und Eulers, ihre Zusammenhänge und ihre Bedeutung. Archive for History of Exact Sciences, I, 122–159.

  • Kline, M. (1999). Storia del pensiero matematico (Vol. 2). Torino: Einaudi.

    Google Scholar 

  • Konen, H. (1901). Geschichte der Gleichung t2–Du2 = 1. Leipzig: Hirzel.

    Google Scholar 

  • Lagrange, J. L. (1769). Sur la solution de problèmes indéterminés du second degré, in Mémoires de l’Académie royale des Sciences et Belles-Lettres de Berlin, t. XXIII. In Fermat 1891–1922 (II, pp. 375–535).

  • Lagrange, J. L. (1770). Démonstration d’un théorème d’arithmétique. Nouveaux Mémoires de l’Académie royale des Sciences et Belles–Lettres de Berlin. In Fermat 1891–1922 (III, pp. 187–201).

  • Lagrange, J. L. (1774). Additions aux Eléments d’Algèbre d’Euler. In., 7, pp. 3-180. See also: Euler, OO, 1, I, pp. 499–651 with the title Additions à l’analyse indéterminée.

  • Lagrange, J. L. (1777). Sur quelques problèmes de l’analyse de Diophante. Nouveaux Mémoires de l’Académie royale des Sciences et Belles–Lettres de Berlin. In Fermat 1891–1922 (IV, pp. 375–398).

  • Lagrange, J. L. (1870–1873). Œuvres de Lagrange. Seconde édition. Paris : Courcier (ed), XIV vols. (in X) Paris: Gauthier–Villars.

  • Lagrange, J. L. ([1773] 1775). Recherches d’arithmétique. Nouveaux Mémoires de l’Académie royale des Sciences et Belles-Lettres de Berlin”. In Fermat 1891–1922 (III, pp. 693–758, 759–795) [first and second part respectively].

  • Lemmermeyer, F. (2003). Higher descent on Pell conics. I. From Legendre to Selmer. http://www.fen.bilkent.edu.tr/~franz/publ/pell-a-pdf.

  • Lucas, E. (1891). Théorie des nombres. Paris: Gauthier-Villars.

    Google Scholar 

  • Macys, J. J. (2007). On Euler’s hypothetical proof. Mathematical Notes, 82(3), 352–356.

    Article  Google Scholar 

  • Mahoney, M. S. ([1973] 1994). The mathematical career of Pierre de Fermat 1601–1665. Princeton: Princeton University Press.

  • OO Euler, L. ([1911, 1913, 1915, 1917] 1941). Opera Omnia, Series 1. Vols. I, X, II, III, IV. Leipzig and Berlin: Teubner.

  • Pieper, H. (1993). On Euler’s contribution to the four–squares theorem. Historia Mathematica, 20, 12–18.

    Article  Google Scholar 

  • Pisano, R., Agassi, J., & Drozdova, D. (Eds.). (2017). Hypothesis and perspective in history and philosophy of science. Homage to Alexandre Koyré 1892–1964. Dordrecht: Springer.

    Google Scholar 

  • Pisano, R., Bussotti, P. (2013). On popularization of scientific education in Italy between 12th and 16th centuries. In Problems of education in the 21st century (Vol. 57, pp. 90–101).

  • Pisano, R., & Bussotti, P. (2016). A Newtonian tale details on notes and proofs in Geneva Edition of Newton’s Principia. Bulletin of the British Society for the History of Mathematics, 31(3), 160–178.

    Article  Google Scholar 

  • Pisano, R., & Bussotti, P. (2017). On the conceptualization of force in Johannes Kepler’s corpus: an interplay between physics/mathematics and metaphysics. Pisano, Agassi and Drozdova 2017, 295–346.

    Google Scholar 

  • Pisano, R., & Capecchi, D. (2013). Conceptual and mathematical structures of mechanical science in the western civilization around 18th century. Almagest, 4(2), 86–121.

    Article  Google Scholar 

  • Pisano, R., & Capecchi, D. (2015). Tartaglia’s science weights. Mechanics in XVI century. Selections from Quesiti et inventioni diverse. Dordrecht: Springer.

    Google Scholar 

  • Pisano, R., & Gaudiello, I. (2009). Continuity and discontinuity. An epistemological inquiry based on the use of categories in history of science. Polish Academy of Sciences. Organon, 41, 245–265.

    Google Scholar 

  • Piyadasa, R. A. D. (2010). Method of infinite descent and proof of Fermat’s last theorem for n = 3. Canadian Journal of Computing in Mathematics, Natural Sciences, Engineering and Medicine, 1(6), 181–186.

    Google Scholar 

  • Ribenboim, P. (1979). 13 Lectures on Fermat’s last theorem. New York: Springer.

    Book  Google Scholar 

  • Selenius, C. O. (1963). Kettenbruchtheoretische Erklärung der zyklischen Methode zur Lösung der Bhashare–Pell- Gleichung. Acta Academiae Aboensis. Mathematica et Physica, XXIII, 3–44.

    Google Scholar 

  • Shirali, S. A. (2003). Infinite descent—but not into Hell. Resonance, 2, 42–55.

    Article  Google Scholar 

  • Smith, S. T. (1992). Quadratic residues and x3 + y3 = z3 in models of IE1 and IE2. Notre Dame Journal of formal logic, 34.

  • Tat-Wing, L. (2005). The method of infinite descent. Mathematical Excalibur, 10(11), 1–2, 4.

  • Vacca, G. (1927–1928). Sul principio della discesa di Fermat e sulle dimostrazioni dell’esistenza degli irrazionali quadratici. Atti della reale accademia delle scienze di Torino, 63, 241–252.

  • Vandiver, H. S. (1932). On the method of infinite descent in connection with Fermat’s last theorem for regular primes. Commentarii Mathematici Elvetici, 4, 1–8.

    Article  Google Scholar 

  • Weil, A. (1977). Fermat et l’équation de Pell. ΠΡΙΣΜΑΤΑ. Naturwissenschaftgeschichtliche Studien (pp. 441–447). Wiesbaden: Franz Steine Verlag.

  • Weil, A. (1984). Number theory: an approach through history from Hammurapi to Legendre. Boston: Birkhäuser.

    Google Scholar 

  • Wirth, C. P. (2004). Descente infinie + deduction. Logic journal of the IGPL, 12(1), 1–96.

    Article  Google Scholar 

  • Wirth, C. P. (2010). A self–contained and easily accessible discussion of the method of descente infinie and FERMAT’s only explicitly known proof by descente infinie. Seki Working-paper SWP-2006-02. arXiv:0902.3623

Download references

Acknowledgements

We acknowledge Gallica National French Library (BnF) for its kind permission and we address our gratitude to anonymous referees for their valuable remarks, which have been of great help.

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Authors and Affiliations

  1. Udine University, Udine, Italy

    Paolo Bussotti

  2. IEMN, Lille University, Lille, France

    Raffaele Pisano

  3. Affiliated to (2015–2019) School/SCFS, Sydney University, Sydney, Australia

    Raffaele Pisano

  4. Affiliated to CPNSS, LSE, London, UK

    Raffaele Pisano

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  1. Paolo Bussotti
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  2. Raffaele Pisano
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Correspondence to Raffaele Pisano.

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Bussotti, P., Pisano, R. Historical and Foundational Details on the Method of Infinite Descent: Every Prime Number of the Form 4n + 1 is the Sum of Two Squares. Found Sci 25, 671–702 (2020). https://doi.org/10.1007/s10699-019-09642-3

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