Diophantine equation
Category:Number theory\nIn
mathematics,
Diophantine equations are equations of the form
f = 0, where
f is a
polynomial with
integer coefficients in one or several variables which take on integral values. They are named after
Diophantus who studied equations with variables which take on rational values. A traditional name for the study of Diophantine equations is
Diophantine analysis. The questions asked include:
- Are there any solutions?\n*Are there any solutions beyond some that are easily found by inspection?\n*Are there finitely or infinitely many solutions?\n*Can all solutions be found, in theory?\n*Can one in practice compute a full list of solutions?
Such problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles. In 1970, a novel result in
mathematical logic known as
Matiyasevich's theorem showed that it is hopeless to expect a complete theory, in effect settling
Hilbert's tenth problem. The point of view of
Diophantine geometry, which is the application of
algebraic geometry techniques in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to equations having a geometric meaning also.
Examples of Diophantine equations are\n*
ax +
by = 1: See
Bézout's identity.\n*
xn +
yn =
zn: For
n = 2 there are many solutions (
x,
y,
z), the Pythagorean triples. For larger values of
n,
Fermat's last theorem states that no positive integer solutions
x,
y,
z satisfying the above equation exist.\n*
x2 -
n y2 = 1: (
Pell's equation) which is named, mistakenly, after the English mathematician
John Pell. It was studied by
Fermat.
One of the few general approaches is through the
Hasse principle.
Infinite descent is the traditional method, and has been pushed a long way.
The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as
recursively enumerable.
The field of
Diophantine approximation deals with the cases of
Diophantine inequalities: variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds.
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