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Issue No.01 - January-March (2008 vol.30)
pp: 38-49
Leo Corry , Tel Aviv University
This article describes the work of Harry Schultz Vandiver, Derrick Henry Lehmer, and Emma Lehmer on calculations related with proofs of Fermat's last theorem. This story sheds light on ideological and institutional aspects of activity in number theory in the US during the 20th century, and on the incursion of computer-assisted methods into pure fields of mathematical research.
Fermat's last theorem, Lehmer, number theory, SWAC, Vandiver
Leo Corry, "Fermat Meets SWAC: Vandiver, the Lehmers, Computers, and Number Theory", IEEE Annals of the History of Computing, vol.30, no. 1, pp. 38-49, January-March 2008, doi:10.1109/MAHC.2008.6
1. Above all through the success of Simon Singh's best-seller Fermat's Enigma and its associated BBC TV program (produced in collaboration with John Lynch).
2. See L. Corry, "El Teorema de Fermat y sus Historias [Fermat's Theorem and Its Histories]," Gaceta de la Real Sociedad Matemática Española, vol. 9, no. 2, 2006, pp. 387-424 (in Spanish), L. Corry, "Fermat Comes to America: Harry Schultz Vandiver and FLT (1914–1963)," Mathematical Intelligencer, vol. 29, 2007, pp. 30-40.
3. For detailed explanations about the theorems and proofs mentioned in this and the next few paragraphs, as well as references to the original sources, see H.M. Edwards, Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory, Springer 1977.
4. E.E. Kummer, "Einige Satze über die aus den Wurzeln der Gleichung ... [Some Theorems on the Roots of the Equation ...]," Math. Abh. Akad. Wiss. Berlin, 1857, pp. 41-74.
5. M. Ohm, "Etwas über die Bernoulli'schen Zahlen [On the Bernoulli Numbers]," Journal für reine und angewandte Mathematik, [Journal for Pure and Applied Mathematics], (abbreviated hereafter as J. für Math.), vol. 20, 1840, pp. 11-12 (in German).
6. J.C. Adams, "Table of the values of the first sixty-two numbers of Bernoulli," J. für Math., vol. 85, 1878, pp. 269-272, S. Serebrenikoff, "Novyi sposob vychisleniya chisel Bernulli" [A New Method of Computation of Bernoulli Numbers], Zap. Akad. Nauk, Sankt Peterburg (Mémoires of the Imperial Academy of Sciences of St. Petersburg), vol. 19, no. 4, 1906, pp. 1-6 (in Russian).
7. K. Løchte Jensen, "Om talteoretiske Egenskaber ved de Bernoulliske Tal [On Number Theoretical Properties of the Bernoulli Numbers]," Nyt Tidsskrift for Matematik, vol. 26, 1915, pp. 73-83 (in Danish).
8. H.S. Vandiver and G.E. Wahlin, Algebraic Numbers-II. Report of the Committee on Algebraic Numbers, National Research Council, 1928, p. 182.
9. D. Hilbert, The Theory of Algebraic Number Fields, Springer 1998, p. ix, The expression "a minimum of blind calculations" quoted earlier in this same context appears in H. Minkowski, "Peter Gustav Lejeune Dirichlet und seine Bedeutung für die heutige Mathematik" [Peter Gustav Lejeune Dirichlet and his Significance for Contemporary Mathematics], Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 14, 1905, pp. 149-163 (in German).
10. For a complete bibliography, see the chapter on FLT in vol. 2 of L.E. Dickson, History of the Theory of Numbers, Chelsea, 1919. Except for items explicitly listed later, works mentioned in this section are all reported in Dickson's book.
11. W. Meissner (1913), "Über die Teilbarkeit von 2p−2durch das Quadrat der Primzahl p = 1093 [On the Divisibility of 2p−2by the Square of the Number p = 1093]," Berlin-Brandenburgische Akademie der Wissenschaften. Berichte und Abhandlungen, [Berlin-Brandenburg Academy of Sciences. Reports and Treatises], 1913, pp. 663-667 (in German).
12. N.G.W.H. Beeger, "On the Congruence 2p−1≡1 (mod p2) and Fermat's Last Theorem," Messenger of Mathematics, vol. 55, 1925, pp. 17-26.
13. For other uses of this machine, see M. Croarken, Early Scientific Computing in Britain, Clarendon Press, 1990, pp. 13-15.
14. N.G.W.H. Beeger, "On the Congruence 2p−1≡1 (mod p2) and Fermat's Last Theorem," Nieuw Archief voor Wiskunde, vol. 20, 1939, pp. 51-54.
15. "Crelle's Journal" is the J. für Math., founded by August L. Crelle in 1826. H.S. Vandiver, "Extensions of the Criteria of Wieferich and Mirimanoff in Connection with Fermat's Last Theorem," J. für Math., vol. 144, 1914, pp. 314-318.
16. H.S. Vandiver, "On Kummer's Memoir of 1857 Concerning Fermat's Last Theorem," Proc. Nat'l Academy of Science (PNAS), vol. 6, 1920, pp. 266-269, For a detailed account of Vandiver's work on FLT, see Corry, "Fermat Comes to America," 2007.
17. H.S. Vandiver, "On Fermat's Last Theorem," Trans. Am. Mathematical Soc. (AMS), vol. 31, 1929, pp. 613-642.
18. D.N. Lehmer, List of Prime Numbers from 1 to 10,006,721, Carnegie Institution of Washington, 1914.
19. The machine is described in D.H. Lehmer, "A Photo-Electric Number-Sieve," Am. Mathematics Monthly, vol. 40, 1933, pp. 401-406.
20. J. Brillhart, "John Derrick Henry Lehmer," Acta Arithmetica, vol. 62, 1992, pp. 207-213.
21. D. Lehmer to Vandiver, 9 Oct. 1934. The bulk of the correspondence between Vandiver and the Lehmers is kept in the Vandiver Collection, Archives of American Mathematics, Center for American History, The Univ. of Texas at Austin (hereafter cited as HSV). Interesting material is also found at the Emma and Dick Lehmer Archive, Univ. of California, Berkeley (hereafter cited as EHL). Letters are quoted here by permission.
22. D.H. Lehmer, "Lacunary Recurrence Formulas for the Numbers of Bernoulli and Euler," Annals of Mathematics, vol. 36, 1935, pp. 637-648.
23. D. Lehmer to Vandiver, 20 Nov. 1934, HSV.
24. Lehmer, "Lacunary Recurrence Formulas," p. 637.
25. D.H. Lehmer, "An Extension of the Table of Bernoulli Numbers," Duke Mathematical J., vol. 2, 1936, pp. 460-464.
26. Dick Lehmer to Vandiver, 10 Feb. 1936, HSV.
27. See M. Campbell-Kelly et al., eds., The History of Mathematical Tables. From Summer to Spreadsheets, Princeton Univ. Press, 2003.
28. See A.N. Lowan, "The Computational Laboratory of the National Bureau of Standards," Scripta Mathematica, vol. 15, 1949, pp. 33-63, Lehmer is not mentioned in a recent account of the history of the project: D.A. Grier, "Table Making for the Relief of Labour," Campbell-Kelly et al., Mathematical Tables, pp. 265-292.
29. H.S. Vandiver, "On Bernoulli Numbers and Fermat's Last Theorem," Duke Mathematical J., vol. 3, 1937, pp. 569-584.
30. H.S. Vandiver, "On Bernoulli Numbers and Fermat's Last Theorem (Second Paper)," Duke Mathematical J., vol. 5, 1939, pp. 418-427.
31. See H.D. Huskey, "SWAC-Standards Western Automatic Computer," IEEE Annals of the History of Computing, vol. 19, no. 4, 1997, pp. 51-61.
32. G.W. Reitwiesner, "An ENIAC Determination of πand e to more than 2000 Decimal Places," Mathematical Tables and Other Aids to Computation, vol. 4, 1950, pp. 11-15.
33. E. Lehmer, "Number Theory on the SWAC," Proc. Symp. Applied Mathematics, vol. 6, AMS, 1956, pp. 103-108.
34. R. Robinson, "Mersenne and Fermat Numbers," Proc. AMS, vol. 5, 1954, pp. 842-846, on p. 844.
35. E. Lehmer to Vandiver, 7 Mar. 1953, HSV.
36. Vandiver to E. Lehmer, 3 Apr. 1953, HSV.
37. See L. Corry, "Number Crunching vs. Number Theory: Computers and FLT, from Kummer to SWAC, and beyond," Archives for History of Exact Science, (forthcoming).
38. Lehmers to Vandiver, cable, 16 June 1953, EDL.
39. Vandiver to E. Lehmer, 22 Sept. 1953, HSV. It should be said that to this day no proof exists of the infiniteness of the regular primes, but there are good arguments to believe that this is the case. See C.L. Siegel, "Zu zwei Bemerkungken Kummers" [On Two Remarks of Kummer], Gött. Nachr., 1964, pp. 51-62 (in German).
40. Vandiver to E. Lehmer, 5 Oct. 1953, HSV. Quoted verbatim.
41. H.S. Vandiver, D.H. Lehmer, and E. Lehmer, "An Application of High-Speed Computing to Fermat's Last Theorem," PNAS, vol. 40, 1954, pp. 25-33, on p. 33.
42. Vandiver to E. Lehmer, 30 Oct. 1953, HSV.
43. E. Lehmer to Vandiver, 14 Aug. 1954, HSV.
44. See J. Todd, "Numerical Analysis at the National Bureau of Standards," SIAM Rev., vol. 17, 1975, pp. 361-370.
45. H.S. Vandiver, "Examination of Methods of Attack on the Second Case of Fermat's Last Theorem," PNAS, vol. 40, 1954, pp. 732-735, H.S. Vandiver, J.L. Selfridge, and C.A. Nicol, "Proof of Fermat's Last Theorem for All Prime Exponents Less Than 4002," PNAS, vol. 41, 1955, pp. 970-973.
46. See, for example, S.S. Wagstaff, "The Irregular Primes to 125,000," Mathematics of Computation, vol. 32, 1978, pp. 583-591, J.P. Buhler, R.E. Crandall, and R.W. Sompolski, "Irregular Primes to One Million," Mathematics of Computation, vol. 59, 1992, pp. 717-722.
47. J. Buhler et al., "Irregular Primes and Cyclotomic Invariants to 12 Million," J. Symbolic Computing, vol. 31, 2001, pp. 89-96.
48. H.S. Vandiver, "The Rapid Computing Machine as an Instrument in the Discovery of New Relations in the Theory of Numbers," PNAS, vol. 44, 1958, pp. 459-464.
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