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Hermite-Lindemann Theorem

Let alpha_i and A_i be algebraic numbers such that the A_is differ from zero and the alpha_is differ from each other. Then the expression

A_1e^(alpha_1)+A_2e^(alpha_2)+A_3e^(alpha_3)+...

cannot equal zero. The theorem was proved by Hermite (1873) in the special case of the A_is and alpha_is rational integers, and subsequently proved for algebraic numbers by Lindemann in 1882 (Lindemann 1888). The proof was subsequently simplified by Weierstrass (1885) and Gordan (1893).

See also

Algebraic Number, Constant Problem, Four Exponentials Conjecture, Integer Relation, Lindemann-Weierstrass Theorem, Six Exponentials Theorem

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References

Dörrie, H. "The Hermite-Lindemann Transcendence Theorem." §26 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 128-137, 1965.Hermite, C. "Sur la fonction exponentielle." Comptes Rendus Acad. Sci. Paris 77, 18-24, 1873.Gordan, P. "Transcendenz von e und pi." Math. Ann. 43, 222-224, 1893.Lindemann, F. "Über die Ludolph'sche Zahl." Sitzungber. Königl. Preuss. Akad. Wissensch. zu Berlin No. 2, pp. 679-682, 1888.Weber, H. Lehrbuch der Algebra, Vols. I-II. New York: Chelsea, 1902.Weierstrass, K. "Zu Hrn. Lindemann's Abhandlung: 'Über die Ludolph'sche Zahl.' " Sitzungber. Königl. Preuss. Akad. Wissensch. zu Berlin No. 2, pp. 1067-1086, 1885.

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Hermite-Lindemann Theorem

Cite this as:

Weisstein, Eric W. "Hermite-Lindemann Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hermite-LindemannTheorem.html

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