Six exponentials theorem
Encyclopedia
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.
:
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 number
s:
The theorem then says that if λij 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.
and Paul Erdős
from 1944 in which they try to prove that the ratio of consecutive colossally abundant number
s is always prime
. They claimed that Carl Ludwig Siegel
knew of a proof of this special case, but it is not recorded. Using the special case they manage to prove that the ratio of consecutive colossally abundant numbers is always either a prime or a semiprime
.
The theorem was first explicitly stated and proved in its complete form independently by Serge Lang
and Kanakanahalli Ramachandra
in the 1960s.
This theorem implies the six exponentials theorem and in turn is implied by the as yet unproven four exponentials conjecture, which says that in fact one of the first four numbers on this list must be transcendental.
Then xi yj = βij for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2. The six exponentials theorem then follows by setting βij = 0 for every i and j, while the five exponentials theorem follows by setting x3 = γ/x1 and using Baker's theorem
to ensure that the xi are linearly independent.
There is a sharp version of the five exponentials theorem as well, although it as yet unproven so is known as the sharp five exponentials conjecture. This conjecture implies both the sharp six exponentials theorem and the five exponentials theorem, and is stated as follows. Let x1, x2 and y1, y2 be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let α, β11, β12, β21, β22, and γ be six algebraic numbers with γ ≠ 0 such that the following five numbers are algebraic:
Then xi yj = βij for 1 ≤ i, j ≤ 2 and γx2 = αx1.
A consequence of this conjecture that isn't currently known would be the transcendence of eπ², by setting x1 = y1 = β11 = 1, x2 = y2 = iπ, and all the other values in the statement to be zero.
over the algebraic numbers generated by 1 and all logarithms of algebraic numbers, denoted here as L∗. So L∗ is the set of all complex numbers of the form
for some n ≥ 0, where all the βi and αi are algebraic and every branch of the logarithm is considered. The strong six exponentials theorem then says that if x1, x2, and x3 are complex numbers that are linearly independent over the algebraic numbers, and if y1 and y2 are a pair of complex numbers that are also linearly independent over the algebraic numbers then at least one of the six numbers xi yj for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2 is not in L∗. This is stronger than the standard six exponentials theorem which says that one of these six numbers is not simply the logarithm of an algebraic number.
There is also a strong five exponentials conjecture that would imply both the strong six exponentials theorem and the sharp five exponentials conjecture. This conjecture claims that if x1, x2 and y1, y2 are two pairs of complex numbers, with each pair being linearly independent over the algebraic numbers, then at least one of the following five numbers is not in L∗:
All the above conjectures and theorems are consequences of the unproven extension of Baker's theorem, that logarithms of algebraic numbers that are linearly independent over the rational numbers are automatically algebraically independent too. The diagram on the right shows the logical implications between all these results.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically transcendental number theory
Transcendence theory
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
, 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 x1, x2,..., xd are d complex numbers that are linearly independent over the rational numbers, and y1, y2,...,yl 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 transcendentalTranscendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
:
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
Four exponentials conjecture
In mathematics, specifically the field of transcendental number theory, 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...
, 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 number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s:
The theorem then says that if λij 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
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
has rank
Rank (linear algebra)
The column rank of a matrix A is the maximum number of linearly independent column vectors of A. The row rank of a matrix A is the maximum number of linearly independent row vectors of A...
2.
History
A special case of the result where x1, x2, and x3 are logarithms of positive integers, y1 = 1, and y2 is real, was first mentioned in a paper by Leonidas AlaogluLeonidas Alaoglu
Leonidas Alaoglu was a Canadian-American mathematician, most famous for his widely-cited result called Alaoglu's theorem on the weak-star compactness of the closed unit ball in the dual of a normed space, also known as the Banach–Alaoglu theorem.- Life and work :Alaoglu was born in Red Deer,...
and Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
from 1944 in which they try to prove that the ratio of consecutive colossally abundant number
Colossally abundant number
In mathematics, a colossally abundant number is a natural number that, in some rigorous sense, has a lot of divisors...
s is always prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
. They claimed that Carl Ludwig Siegel
Carl Ludwig Siegel
Carl Ludwig Siegel was a mathematician specialising in number theory and celestial mechanics. He was one of the most important mathematicians of the 20th century.-Biography:...
knew of a proof of this special case, but it is not recorded. Using the special case they manage to prove that the ratio of consecutive colossally abundant numbers is always either a prime or a semiprime
Semiprime
In mathematics, a semiprime is a natural number that is the product of two prime numbers. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... ....
.
The theorem was first explicitly stated and proved in its complete form independently by Serge Lang
Serge Lang
Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
and Kanakanahalli Ramachandra
Kanakanahalli Ramachandra
Kanakanahalli Ramachandra was an Indian mathematician working in analytic number theory.-Early career:...
in the 1960s.
Five exponentials theorem
A stronger, related result is the five exponentials theorem, which is as follows. Let x1, x2 and y1, y2 be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let γ be a non-zero algebraic number. Then at least one of the following five numbers is transcendental:This theorem implies the six exponentials theorem and in turn is implied by the as yet unproven four exponentials conjecture, which says that in fact one of the first four numbers on this list must be transcendental.
Sharp six exponentials theorem
Another related result that implies both the six exponentials theorem and the five exponentials theorem is the sharp six exponentials theorem. This theorem is as follows. Let x1, x2, and x3 be complex numbers that are linearly independent over the rational numbers, and let y1 and y2 be a pair of complex numbers that are linearly independent over the rational numbers, and suppose that βij are six algebraic numbers for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2 such that the following six numbers are algebraic:Then xi yj = βij for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2. The six exponentials theorem then follows by setting βij = 0 for every i and j, while the five exponentials theorem follows by setting x3 = γ/x1 and using Baker's theorem
Baker's theorem
In transcendence theory, a mathematical discipline, Baker's theorem gives a lower bound for linear combinations of logarithms of algebraic numbers...
to ensure that the xi are linearly independent.
There is a sharp version of the five exponentials theorem as well, although it as yet unproven so is known as the sharp five exponentials conjecture. This conjecture implies both the sharp six exponentials theorem and the five exponentials theorem, and is stated as follows. Let x1, x2 and y1, y2 be two pairs of complex numbers, with each pair being linearly independent over the rational numbers, and let α, β11, β12, β21, β22, and γ be six algebraic numbers with γ ≠ 0 such that the following five numbers are algebraic:
Then xi yj = βij for 1 ≤ i, j ≤ 2 and γx2 = αx1.
A consequence of this conjecture that isn't currently known would be the transcendence of eπ², by setting x1 = y1 = β11 = 1, x2 = y2 = iπ, and all the other values in the statement to be zero.
Strong six exponentials theorem
A further strengthening of the theorems and conjectures in this area are the strong versions. The strong six exponentials theorem is a result proved by Damien Roy that implies the sharp six exponentials theorem. This result concerns the vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over the algebraic numbers generated by 1 and all logarithms of algebraic numbers, denoted here as L∗. So L∗ is the set of all complex numbers of the form
for some n ≥ 0, where all the βi and αi are algebraic and every branch of the logarithm is considered. The strong six exponentials theorem then says that if x1, x2, and x3 are complex numbers that are linearly independent over the algebraic numbers, and if y1 and y2 are a pair of complex numbers that are also linearly independent over the algebraic numbers then at least one of the six numbers xi yj for 1 ≤ i ≤ 3 and 1 ≤ j ≤ 2 is not in L∗. This is stronger than the standard six exponentials theorem which says that one of these six numbers is not simply the logarithm of an algebraic number.
There is also a strong five exponentials conjecture that would imply both the strong six exponentials theorem and the sharp five exponentials conjecture. This conjecture claims that if x1, x2 and y1, y2 are two pairs of complex numbers, with each pair being linearly independent over the algebraic numbers, then at least one of the following five numbers is not in L∗:
All the above conjectures and theorems are consequences of the unproven extension of Baker's theorem, that logarithms of algebraic numbers that are linearly independent over the rational numbers are automatically algebraically independent too. The diagram on the right shows the logical implications between all these results.