Algebraic equations, or systems of algebraic equations with rational
coefficients, the solutions of which are sought for in integers
or rational numbers. It is usually assumed that the number of
unknowns in Diophantine equations is larger than the number
of equations; thus, they are also known as
indefinite equations.
In modern mathematics the concept of a Diophantine equation is also
applied to algebraic equations the solutions of which are sought for
in the algebraic integers of some algebraic extension of the field
of rational numbers, of the field of
-adic
numbers, etc.
The study of Diophantine equations is on the
border-line between number theory and algebraic geometry (cf.
Diophantine geometry).
Finding solutions of equations in integers is one of the oldest
mathematical problems. As early as the beginning of the
second millennium B.C.
ancient Babylonians succeeded in solving systems of equations
with two unknowns. This branch of mathematics flourished to the
greatest extent in Ancient Greece. The principal source is
Aritmetika
by
Diophantus
(probably the
3rd century A.D.),
which contains
different types of equations and systems. In this book,
Diophantus
(hence the name
"Diophantine equations" )
anticipated a number of
methods for the study of equations of the second and
third degrees which were only fully developed in the
19th century
[1].
The creation of the theory of rational numbers by the
scientists of Ancient Greece led to the study of
rational solutions of indefinite equations.
This point of view is systematically followed by
Diophantus
in his book. Even though his work contains solutions
of specific Diophantine equations only, there is reason to believe
that he was also familiar with a few general methods.
The study of Diophantine equations usually involves major
difficulties. Moreover, it is possible to specify, explicitly, polynomials
with integer coefficients such that no algorithm exists by which
it would be possible to tell, for any given
,
whether the equation
is solvable for
(cf.
Diophantine equations, solvability problem of).
Examples of such polynomials may be explicitly
written down; no exhaustive description of their solutions can be given (if the
Church thesis
is accepted).
The simplest Diophantine equation
where
and
are relatively prime integers, has infinitely many solutions (if
form a solution, then the pair of numbers
and
,
where
is an arbitrary integer, will also be a
solution). Another example of a Diophantine equation is
Positive integral solutions of this equation represent the lengths of the small sides
and of the hypotenuse
of right-angled triangles with integral side lengths; these numbers are known as
Pythagorean numbers.
All triplets of relatively prime Pythagorean numbers are given by the formulas
where
and
are relatively prime integers
(
).
Diophantus
in his
Aritmetika
deals with the search for rational (not necessarily
integral) solutions of special types of Diophantine equations. The
general theory of solving of Diophantine equations of the first
degree was developed by
C.G. Bachet
in the
17th century;
for more details on this subject see
Linear equation.
P. Fermat,
J. Wallis,
L. Euler,
J.L. Lagrange,
and
C.F. Gauss
in
the early
19th century
mainly studied Diophantine equations of the form
where
,
,
,
,
,
and
are integers, i.e. general inhomogeneous equations of
the second degree with two unknowns. Lagrange used
continued fractions
in his study of general inhomogeneous Diophantine equations of the second
degree with two unknowns. Gauss developed the general theory of
quadratic forms,
which is the basis of solving certain types of Diophantine equations.
In studies on Diophantine equations of degrees higher than two significant
success was attained only in the
20th century.
It
was established by
A. Thue
that the Diophantine equation
where
,
are integers, and the polynomial
is irreducible in the field of rational numbers, cannot
have an infinite number of integer solutions. However,
Thue's method
fails to yield either a bound on the solutions or on
their number.
A. Baker
obtained effective theorems giving bounds on
solutions of certain equations of this kind.
B.N. Delone
proposed
another method of investigation, which is applicable to a narrower class
of Diophantine equations, but which yields a bound for the
number of solutions. In particular, Diophantine equations of the form
are fully solvable by this method.
The theory of Diophantine equations has many directions.
Thus, a well-known problem in this theory is
Fermat's problem
— the
hypothesis according to which there are no
non-trivial solutions of the Diophantine equation
if
.
The study of integer solutions of equation
(1)
is a natural generalization of the problem of
Pythagorean triplets.
Euler obtained a positive solution of Fermat's problem for
.
Owing to this result, Fermat's problem is reduced to the proof
of the absence of non-zero integer solutions of equation
(1)
if
is an odd prime. At the time of writing
(
1988)
the study
concerned with solving
(1)
has not been completed. The difficulties involved
in solving it are due to the fact that prime factorization
in the ring of algebraic integers is not unique. The theory
of divisors in rings of algebraic integers makes it possible to
confirm the validity of Fermat's theorem for many classes of prime exponents
.
The arithmetic of rings of algebraic integers is also utilized
in many other problems in Diophantine equations. For instance, such methods
were applied in a detailed solution of an equation of the form
where
is the norm of the algebraic number
,
and integral rational numbers
which satisfy equation
(2)
are to be found.
Equations of this class include, in particular, the
Pell equation
.
Depending on the values of
which appear in
(2),
these equations are subdivided into
two types. The first type — the so-called
complete forms
—
comprises equations in which among the
there are
linearly independent numbers over the field of rational numbers
,
where
is the degree of the algebraic number field
over
.
Incomplete forms
are those in which the maximum number of linearly independent numbers
is less than
.
The case of complete forms is simpler and its study
has now, in principle, been completed. It is possible, for
example, to describe all solutions of any complete form
[2].
The second type — the incomplete forms — is more
complicated, and the development of its theory is still
(1988)
far
from being completed. Such equations are studied with the aid of
Diophantine approximations.
They include the equation
where
is an irreducible homogeneous polynomial of degree
.
This equation may be written as
where
are all the roots of the polynomial
.
The existence of an infinite sequence of integral solutions of
equation
(3)
would lead to relationships of the form
for some
.
Without loss of generality, one may assume that
.
Accordingly, if
is sufficiently large, inequality
(4)
will be in contradiction with the
Thue–Siegel–Roth theorem,
from which follows that the equation
,
where
is an irreducible form of degree three or
higher, cannot have an infinite number of solutions.
Equations such as
(2)
constitute a fairly narrow class
among all Diophantine equations. For instance,
their simple appearance notwithstanding, the equations
and
are not in this class. The study of the solutions of equation
(6)
is a fairly thoroughly investigated branch of Diophantine equations — the
representation of numbers by quadratic forms.
The
Lagrange theorem
states that
(6)
is solvable for all natural
.
Any natural number not representable in the form
,
where
and
are non-negative integers, can be represented as a
sum of three squares
(Gauss' theorem).
Criteria are
known for the existence of rational or integral solutions of equations of the form
where
is a quadratic form with integer coefficients. Thus, according to
Minkowski–Hasse theorem,
the equation
where
and
are rational, has a rational solution if and only
if it is solvable in real numbers and in
-adic
numbers for each prime number
.
The representation of numbers by arbitrary forms of the third degree or
higher has been studied to a lesser extent, because of inherent
difficulties. One of the principal methods of study in the representation
of numbers by forms of higher degree is the method of trigonometric sums (cf.
Trigonometric sums, method of).
In this method the number of solutions of the equation
is explicitly written out in terms of a Fourier integral, after which the
circle method
is employed to express the number of solutions of the equation in terms
of the number of solutions of the corresponding congruences. The method
of trigonometric sums depends less than do other
methods on the algebraic peculiarities of the equation.
There exists a large number of specific Diophantine
equations which are solvable by elementary methods
[5].