In mathematics, a Diophantine equation is an equation between two polynomials with integer coefficients with any number of unknowns. A Diophantine problem is given as a Diophantine equation, whose solutions are the possible assignments of integers for the unknowns for which the equation is satisfied.
The word Diophantine refers to the Greek mathematician of the third century A.D., Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems Diophantus initiated is now called Diophantine analysis.
A linear Diophantine equation is an equation between two sums of monomials of degree zero or one.
Examples of Diophantine equations
ax + by = 1: See Bézout's identity; this is a linear Diophantine.
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.
x2 - n y2 = 1: (Pell's equation) which is named, mistakenly, after the English mathematician John Pell. It was studied by Fermat.
, where and : These are the Thue equations, and are, in general, solvable.
The questions asked in Diophantine analysis include:
- Are there any solutions?
- Are there any solutions beyond some that are easily found by 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?
Hilbert's tenth problem
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. In 1900, in recognition of their depth, Hilbert proposed the solvability of all Diophantine problems as the tenth of his celebrated problems. In 1970, a novel result in mathematical logic known as Matiyasevich's theorem settled the problem negatively: in general Diophantine problems are unsolvable.
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 also having a geometric meaning.
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.