Editing Diophantine equation (section)

==Diophantine analysis==

=== Typical questions ===

The questions asked in Diophantine analysis include:

#Are there any solutions?
#Are there any solutions beyond some that are easily found by [[List of mathematical jargon#Proof techniques|inspection]]?
#Are there finitely or infinitely many solutions?
#Can all solutions be found in theory?
#Can one in practice compute a full list of solutions?

These traditional problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles.

=== Typical problem ===
The given information is that a father's age is 1 less than twice that of his son, and that the digits {{mvar|AB}} making up the father's age are reversed in the son's age (i.e. {{mvar|BA}}). This leads to the equation {{math|10''A'' + ''B'' {{=}} 2(10''B'' + ''A'') − 1}}, thus {{math|19''B'' − 8''A'' {{=}} 1}}. Inspection gives the result {{math|''A'' {{=}} 7}}, {{math|''B'' {{=}} 3}}, and thus {{mvar|AB}} equals 73 years and {{mvar|BA}} equals 37 years. One may easily show that there is not any other solution with {{mvar|A}} and {{mvar|B}} positive integers less than 10.

Many well known puzzles in the field of [[recreational mathematics]] lead to diophantine equations.  Examples include the [[cannonball problem]], [[Archimedes's cattle problem]] and [[the monkey and the coconuts]].

===17th and 18th centuries===
In 1637, [[Pierre de Fermat]] scribbled on the margin of his copy of ''[[Arithmetica]]'': "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers." Stated in more modern language, "The equation {{math|1=''a{{sup|n}}'' + ''b{{sup|n}}'' = ''c{{sup|n}}''}} has no solutions for any {{mvar|n}} higher than 2." Following this, he wrote: "I have discovered a truly marvelous proof of this proposition, which this margin is too narrow to contain." Such a proof eluded mathematicians for centuries, however, and as such his statement became famous as [[Fermat's Last Theorem]]. It was not until 1995 that it was proven by the British mathematician [[Andrew Wiles]].

In 1657, Fermat attempted to solve the Diophantine equation {{math|61''x''<sup>2</sup> + 1 {{=}} ''y''<sup>2</sup>}} (solved by [[Brahmagupta]] over 1000 years earlier). The equation was eventually solved by [[Euler]] in the early 18th century, who also solved a number of other Diophantine equations. The smallest solution of this equation in positive integers is {{math|''x'' {{=}} 226153980}}, {{math|''y'' {{=}} 1766319049}} (see [[Chakravala method]]).

===Hilbert's tenth problem===
{{main article|Hilbert's tenth problem}}
In 1900, [[David Hilbert]] proposed the solvability of all Diophantine equations as [[Hilbert's tenth problem|the tenth]] of his [[Hilbert's problems|fundamental problems]]. In 1970, [[Yuri Matiyasevich]] solved it negatively, building on work of [[Julia Robinson]], [[Martin Davis (mathematician)|Martin Davis]], and [[Hilary Putnam]] to prove that a general [[algorithm]] for solving all Diophantine equations [[proof of impossibility|cannot exist]].

===Diophantine geometry===
[[Diophantine geometry]], is the application of techniques from [[algebraic geometry]] which considers equations that also have a geometric meaning. The central idea of Diophantine geometry is that of a [[rational point]], namely a solution to a polynomial equation or a [[system of polynomial equations]], which is a vector in a prescribed [[field (mathematics)|field]] {{mvar|K}}, when {{mvar|K}} is ''not''  [[algebraically closed]].

===Modern research===
The oldest general method for solving a Diophantine equation{{mdash}}or for proving that there is no solution{{mdash}} is the method of [[infinite descent]], which was introduced by [[Pierre de Fermat]]. Another general method is the [[Hasse principle]] that uses [[modular arithmetic]] modulo all prime numbers for finding the solutions. Despite many improvements these methods cannot solve most Diophantine equations.

The difficulty of solving Diophantine equations is illustrated by [[Hilbert's tenth problem]], which was  set in 1900 by [[David Hilbert]]; it was to find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. [[Matiyasevich's theorem]] implies that such an algorithm cannot exist.

During the 20th century, a new approach has been deeply explored, consisting of using [[algebraic geometry]]. In fact, a Diophantine equation can be viewed as the equation of an [[hypersurface]], and the solutions of the equation are the points of the hypersurface that have integer coordinates.

This approach led eventually to the [[Wiles's proof of Fermat's Last Theorem|proof by Andrew Wiles]] in 1994 of [[Fermat's Last Theorem]], stated without proof around 1637. This is another illustration of the difficulty of solving Diophantine equations.

===Infinite Diophantine equations===
An example of an infinite Diophantine equation is:
<math display=block>n = a^2 + 2b^2 + 3c^2 + 4d^2 + 5e^2 + \cdots,</math>
which can be expressed as "How many ways can a given integer {{mvar|n}} be written as the sum of a square plus twice a square plus thrice a square and so on?" The number of ways this can be done for each {{mvar|n}} forms an integer sequence. Infinite Diophantine equations are related to [[theta functions]] and infinite dimensional lattices. This equation always has a solution for any positive {{mvar|n}}.<ref>{{cite web | url=https://oeis.org/A320067 | title=A320067 - Oeis }}</ref> Compare this to:
<math display=block>n = a^2 + 4b^2 + 9c^2 + 16d^2 + 25e^2 + \cdots,</math>
which does not always have a solution for positive {{mvar|n}}.