In this way, the statement of the six exponentials theorem can be generalized to an arbitrary commutative group variety over the field of algebraic numbers.
2.
The related six exponentials theorem was first explicitly mentioned in the 1960s by Serge Lang and Kanakanahalli Ramachandra, and both also explicitly conjecture the above result.
3.
Apart from prime number theory, he made substantial contributions to the theory of transcendental number theory, in which he is known for his proof of the six exponentials theorem, achieved independently of Serge Lang.
4.
This conjecture implies both the sharp six exponentials theorem, which requires a third " x " value, and the as yet unproven sharp five exponentials conjecture that requires a further exponential to be algebraic in its hypotheses.
5.
Indeed, after proving the six exponentials theorem Lang mentions the difficulty in dropping the number of exponents from six to four & mdash; the proof used for six exponentials just misses when one tries to apply it to four.
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