![Y^x = x^y](https://upload.wikimedia.org/wikipedia/commons/f/f4/Y%5Ex_%3D_x%5Ey.svg)
In general,
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
fails to be
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
. However, the
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
holds in special cases, such as
History
The equation
is mentioned in a letter of
Bernoulli Bernoulli can refer to:
People
*Bernoulli family of 17th and 18th century Swiss mathematicians:
** Daniel Bernoulli (1700–1782), developer of Bernoulli's principle
**Jacob Bernoulli (1654–1705), also known as Jacques, after whom Bernoulli numbe ...
to
Goldbach (29 June 1728
). The letter contains a statement that when
the only solutions in
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s are
and
although there are infinitely many solutions in
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s, such as
and
.
The reply by Goldbach (31 January 1729
) contains general solution of the equation, obtained by substituting
A similar solution was found by
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
.
J. van Hengel pointed out that if
are positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s with
, then
therefore it is enough to consider possibilities
and
in order to find solutions in natural numbers.
The problem was discussed in a number of publications.
In 1960, the equation was among the questions on the
William Lowell Putnam Competition
The William Lowell Putnam Mathematical Competition, often abbreviated to Putnam Competition, is an annual mathematics competition for undergraduate college students enrolled at institutions of higher learning in the United States and Canada (regar ...
,
which prompted Alvin Hausner to extend results to
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s.
Positive real solutions
:''Main source:''
A general solution to
is obtained by noting that the positive real quadrant can be 'covered' by the intersection of the two equations
and
(
). Requiring that some points also satisfy the equation
, means that
, and by comparing exponents,
. Thus, the 'covering' equations mean that
, and by exponentiating both sides by
(
),
, and
. The case of
corresponds to the solution
. The full solution thus is
.
Based on the above solution, the derivative
is 1 for the
pairs on the line
, and for the other
pairs can be found by
, which straightforward calculus gives as
for
and
.
The following treatment explores some special cases and notes linkages to other mathematical concepts.
An
infinite
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
* Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
set of trivial solutions in positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s is given by
Nontrivial solutions can be written explicitly as
: