The four exponentials conjecture, in transcendental number theory, states that given the right conditions on the exponents, at least one of four exponentials must be transcendental. Along with two related, stronger conjectures, it sits at the top of a hierarchy of conjectures and theorems concerning the arithmetic nature of certain values of the exponential function. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementIf x1, x2 and y1, y2 are two pairs of complex numbers, with each pair linearly independent over the rational numbers, then at least one of the four numbers e to the power of x1 times y1, e to the power of x1 times y2, e to the power of x2 times y1, and e to the power of x2 times y2 is transcendental. 1 Proposed YearAtle Selberg considered the conjecture informally in the early 1940s without stating it formally; Leonidas Alaoglu and Paul Erdos mentioned a special case in a 1944 paper crediting Carl Ludwig Siegel; the first formal statement in print was Theodor Schneider's, in 1957. Progress Toward ResolutionThe related six exponentials theorem, a weaker statement with six numbers instead of four, has been proved. The four exponentials conjecture itself remains open; the technique used to prove the six exponentials case falls just short when applied to four. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Four Exponentials Conjecture (Wikipedia)
Sources
1. Four Exponentials Conjecture (Wikipedia)
Wikimedia FoundationLead section
the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials
Statement section, opening sentence
If x1, x2 and y1, y2 are two pairs of complex numbers, with each pair being linearly independent over the rational numbers, then at least one of the following four numbers is transcendental:
History section, Schneider sentence
An equivalent statement was first mentioned in print by Theodor Schneider who set it as the first of eight important, open problems in transcendental number theory in 1957.
History section, Lang sentence
Indeed, after proving the six exponentials theorem Lang mentions the difficulty in dropping the number of exponents from six to four, the proof used for six exponentials "just misses" when one tries to apply it to four.
In Branch: Number Theory, Lead sentence
ematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the
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