In general,
exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
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. Perhaps most familiar as a pr ...
. However, the
equation
In mathematics, an equation is a mathematical 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 ...
has an infinity of solutions, consisting of the line and a
smooth curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
intersecting the line at , where is
Euler's number
The number is a mathematical constant approximately equal to 2.71828 that is the base of the natural logarithm and exponential function. It is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can ...
. The only integer solution that is on the curve is .
History
The equation
is mentioned in a letter of
Bernoulli to
Goldbach (29 June 1728
). The letter contains a statement that when
the only solutions in
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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 (for example,
The set of all ...
s, such as
and
.
The reply by Goldbach (31 January 1729
) contains a 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 polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
.
J. van Hengel pointed out that if
are positive
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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,
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 ...
s.
Positive real solutions
:''Main source:''
Explicit form
An
infinite 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s is given by
Nontrivial solutions can be written explicitly using the
Lambert ''W'' function. The idea is to write the equation as
and try to match
and
by multiplying and raising both sides by the same value. Then apply the definition of the Lambert ''W'' function
to isolate the desired variable.
:
:
:
Where in the last step we used the
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
.
Here we split the solution into the two branches of the Lambert ''W'' function and focus on each interval of interest, applying the identities:
:
*
:
:
:
*
:
:
:
*
:
:
:
*
:
:
:
Hence the non-trivial solutions are:
Parametric form
Nontrivial solutions can be more easily found by assuming
and letting
Then
:
Raising both sides to the power
and dividing by
, we get
:
Then nontrivial solutions in positive real numbers are expressed as the
parametric equation
In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters.
In the case ...
The full solution thus is
Based on the above solution, the derivative
is
for the
pairs on the line
and for the other
pairs can be found by
which straightforward calculus gives as:
:
for
and
Setting
or
generates the nontrivial solution in positive integers,
Other pairs consisting of
algebraic number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
s exist, such as
and
, as well as
and
.
The parameterization above leads to a geometric property of this curve. It can be shown that
describes the
isocline curve where power functions of the form
have slope
for some positive real choice of
. For example,
has a slope of
at
which is also a point on the curve
The trivial and non-trivial solutions intersect when
. The equations above cannot be evaluated directly at
, but we can take the
limit as
. This is most conveniently done by substituting
and letting
, so
:
Thus, the line
and the curve for
intersect at .
As
, the nontrivial solution asymptotes to the line
. A more complete asymptotic form is
:
Other real solutions
An infinite set of discrete real solutions with at least one of
and
negative also exist. These are provided by the above parameterization when the values generated are real. For example,
,
is a solution (using the real cube root of
). Similarly an infinite set of discrete solutions is given by the trivial solution
for
when
is real; for example
.
Similar graphs
Equation
The equation
produces a
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...
where the line and curve intersect at
. The curve also terminates at (0, 1) and (1, 0), instead of continuing on to infinity.
The curved section can be written explicitly as