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In mathematics, specifically transcendental number theory, the six exponentials theorem is a result that, given the right conditions on the exponents, guarantees the transcendence of at least one of a set of exponentials. ==Statement== If ''x''1, ''x''2,..., ''x''''d'' are ''d'' complex numbers that are linearly independent over the rational numbers, and ''y''1, ''y''2,...,''y''''l'' are ''l'' complex numbers that are also linearly independent over the rational numbers, and if ''dl'' > ''d'' + ''l'', then at least one of the following ''dl'' numbers is transcendental: : The most interesting case is when ''d'' = 3 and ''l'' = 2, in which case there are six exponentials, hence the name of the result. The theorem is weaker than the related but thus far unproved four exponentials conjecture, whereby the strict inequality ''dl'' > ''d'' + ''l'' is replaced with ''dl'' ≥ ''d'' + ''l'', thus allowing ''d'' = ''l'' = 2. The theorem can be stated in terms of logarithms by introducing the set ''L'' of logarithms of algebraic numbers: : The theorem then says that if λ''ij'' are elements of ''L'' for ''i'' = 1, 2 and ''j'' = 1, 2, 3, such that λ11, λ12, and λ13 are linearly independent over the rational numbers, and λ11 and λ21 are also linearly independent over the rational numbers, then the matrix : has rank 2. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「six exponentials theorem」の詳細全文を読む スポンサード リンク
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