Abstract
In 1832, Dirichlet completed two studies published in Crelle’s Journal. The first was prompted by Gauss’s expansion of the study of residues to complex numbers [Gauss 1832], and the second was the proof of Fermat’s Last Theorem for the case \(n=14\). These were followed by a total of seventeen publications appearing prior to the summer of 1839, of which twelve related to presentations given to the Akademie.
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Notes
- 1.
Gauss 1832b, art. 67.
- 2.
Edwards 1977a, Chapters 3 and 4; also see Edwards 1975 and 1977b.
- 3.
Lagrange 1773/1775; reissued in Lagrange’s Oeuvres 3:789.
- 4.
D.A., art. 131.
- 5.
Except for Lagrange 1775, Dirichlet did not give specific references in this discussion. The reader may wish to note the following: Legendre 1785; Gauss 1801 (reciprocity proofs 1 and 2: D.A. art. 131ff. and art. 262); Gauss 1808a \(=\) Gauss Werke 2:3 (proof 4); Gauss 1808b \(=\) Gauss Werke 2:9 (proof 5); Gauss 1817 \(=\) Gauss Werke 2:51 and 55 (proofs 6 and 7); Analysis Residuorum \(=\) Gauss Werke 2:234 (proof 3, unpublished in Dirichlet’s lifetime). The numbering of Gauss’s proofs in Bachmann 1872:103 differs, as Bachmann does not include proof 3. In addition, Weil 1983 contains several careful discussions concerning Euler’s and Legendre’s work on quadratic reciprocity.
- 6.
In none of his publications until after his return from Italy in 1845 did Dirichlet use the term “Pell’s equation” but always either spelled out the equation or occasionally referred to it as “Fermat’s equation” when the context is unmistakeable. Dirichlet’s publications after 1845 include Pell’s name more than once, presumably a sign that he had become more conscious of Euler’s consistent use of the term and that it was now too firmly associated with that important equation to be disregarded. Legendre, too, had resisted the reference to Pell; his reason had a more nationalistic basis; however, for in the preface to Legendre 1808 he related it to the competition between England and France. Gauss also had resisted using the expression; see the D.A. art. 202.
- 7.
Dirichlet’s account varies slightly from the summary given by Gauss in art. 202 of the D.A. Among other references, note Dickson 1919–23 (2005) 2: Chapter XII.
- 8.
Lagrange Oeuvres 1:671–731.
- 9.
For references to Lagrange’s significant development and publications of this topic between 1761 and 1771, see Weil 1983:233.
- 10.
Writing in French, Dirichlet here used Legendre’s term.
- 11.
Gauss 1811. Gauss had presented it to Göttingen’s Royal Society in 1808, but, as in Berlin, there was a delay in publication; it only appeared three years later.
- 12.
The influence of this memoir is suggested in Chapter 2 of [Patterson 2010]. Patterson, commenting on the statement in [Davenport 2000] that Dirichlet’s method “is probably the most satisfactory of all that are known,” added that it is also the one least frequently reproduced. This is not surprising if one follows the entire long argument, which has been considerably condensed in our lengthy outline.
- 13.
Although described as an “excerpt,” 1837f is actually an only slightly modified summary of the memoir 1837a printed in the Abhandlungen.
- 14.
Werke 2:351.
- 15.
The full proof in 1839a, while rigorous, is very long and, because of its detail, appears complex. We provide a greatly abbreviated outline. The reader interested in proof details may wish, in addition to consulting the memoir itself and Dirichlet’s later expansion of the theorem to quadratic forms, to take note of subsequent smoother but equally rigorous presentations of the 1837 argument. Supplement VI in Dedekind’s editions of Dirichlet’s Lectures on Number Theory, except for some reorganization and Dedekind’s customary increased clarity, comes closest to Dirichlet’s 1837 proof; see Chapter 16.3. Among more recent discussion and proofs, Chapter 1 of [Davenport 2000] also includes an indication of the different approaches Dirichlet took to the problem.
- 16.
Werke 1:309.
- 17.
Werke 1:135.
- 18.
For a more detailed treatment of trigonometric (Fourier) series with related nineteenth-century references, Whittaker & Watson (1927) 1962, Chapter IX, is still useful. Among the numerous nineteenth-century memoirs, notably Riemann 1867, attempting to refine the concept of representing an arbitrary function, note Lipschitz 1864 and du Bois-Reymond 1873. A sound early overview was provided by Sachse 1880; it was subsequently superseded by Dugac 1981.
- 19.
Werke 1:155.
- 20.
Werke 1:159–60.
- 21.
He also reviewed an approach given by Poisson, noting its insufficiency.
- 22.
Werke 1:305–6.
- 23.
Heine 1861:266. Aside from studying the entire discussion in Werke 1:285–306, the reader wishing to work through the details of Dirichlet’s rather lengthy argument may wish to consult the references in Heine 1861 just listed, and note several remarks, with allusions to set theory, in Dauben 1979:10–11.
- 24.
Lagrange Oeuvres 3:788; Lagrange later provided a full proof.
- 25.
Werke 1:349–50.
- 26.
See Sect. 13.5.
- 27.
Euler 1748.
- 28.
His promised extension is 1839–40; see Chap. 11 for this and the later 1842b.
- 29.
1838b; see Werke 1:360.
- 30.
Werke 1:368–69.
- 31.
D.A. art. 287, the penultimate paragraph.
- 32.
When editing this memoir for the Werke, Kronecker called attention to Dirichlet’s having added a handwritten note in the copy of the memoir he sent to Gauss, pointing out that Legendre’s proof is only exact in the first term, the true “expression-limite” being \(\sum {\frac{1}{\log (n)}}\). See Dirichlet’s Werke 1:372n.
- 33.
Dirichlet referred to Euler’s Differential Calculus p. 444, for the constant. [See Institutiones Calculi Differentialis, Opera Omnia Ser. 1, vol. 10.]
- 34.
See Werke 1:373.
- 35.
Werke 1:373–74.
- 36.
He based this remark on a statement of Monge later proved by Chasles. Dirichlet may have learned of Monge’s proposition while studying with Lacroix and Hachette in the 1820s.
- 37.
See Dirichlet–Arendt 1904, Section 8; also, Dirichlet–Meyer 1871: art. 173ff.
- 38.
See, for example, the succinct reference in Courant–Hilbert 1931:69.
- 39.
This merging exemplifies that combinatorial activity of which the psychologist Jerome Bruner wrote while describing creativity. See Bruner 1962.
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Merzbach, U.C. (2018). Publications: Autumn 1832–Spring 1839. In: Dirichlet. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-01073-7_9
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