Videnskabshistorisk Selskab

David E. RoweDavid E. Rowe,
Department of Physics, Mathematics and Computer Science, Mainz University:

On Proofs of the Transcendence of e and π as Background to Hilbert's Seventh Paris Problem

eπi + 1 = 0
Hilbert’s seventh Paris problem concerns proving conjectures about the irrationality or transcendence of certain types of numbers. The inspiration for such conjectures came from two famous results obtained earlier: Hermite’s proof of the transcendence of e in 1873 and its extension in 1882 by Lindemann to the case of  π , which he obtained by showing that the equation ez+1=0  has no algebraic solutions. In 1893 Hilbert managed to prove both results in just four pages, a feat that led to further simplifications by Hurwitz and Gordan. The following year, Klein taught a course for future Gymnasium teachers in which he gave a detailed elementary proof published in 1895 and in numerous translations thereafter. After this, few people probably ever again read Lindemann’s original paper, while the curious and rather amusing story behind it went untold. Soon after it appeared, Lindemann became famous for having resolved (in the negative sense) the ancient problem of squaring the circle, which Greek geometers had only been able to solve by means of transcendental curves. After describing this classical background, we turn to the events of the 1880s and 90s, the entangled careers of Klein, Lindemann, and Hilbert, and to a reconsideration of these events in the light of Hilbert’s seventh problem and its place in the history of mathematics.

tirsdag, den 8. marts 2011, kl. 17.00

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