## Diophantine equationAn equation that has integer coefficients and for which integer solutions are required. Such equations are named after Diophantus. The best known examples are those from Pythagoras's theorem, a^{2} = b^{2} +
c^{2}, when a, b, and c are all required
to be whole numbers – a so-called Pythagorean
triplet. Despite their simple appearance Diophantine equations can be
fantastically difficult to solve. A notorious example comes from Fermat's
last theorem (recently solved), a = ^{n}b
+ ^{n}c for ^{n}n > 2. To give a specific example, suppose we want to find integer values for x
and y such thatx^{2} = 1620y^{2} + 1.
A trial and error approach using a computer would quickly find the solution: y = 4, x = 161. However, just a slight change to the equation
to make it x^{2} = 1621y^{2} + 1
would leave the trial and error method floundering, even with the resources of the most powerful computers on Earth. The smallest integer solution to this innocent looking formula involves a y-value that is on the
order of a thousand trillion trillion trillion trillion trillion trillion!
One of the challenges (the tenth one) that David Hilbert threw down to 20th-century mathematicians in his famous list was to find a general method for solving equations of this type. In 1970, however, the Russian mathematician Yuri Matiyasevich showed that there is no general algorithm for determining whether a particular Diophantine equation is soluble: the problem is undecidable. ^{1, 2}## References- Matijasevic, Yu. V. "Solution of the Tenth Problem of Hilbert."
*Mat. Lapok*, 21: 83- 87 (1970). - Matijasevic, Yu. V.
*Hilbert's Tenth Problem*. Cambridge, Mass.: MIT Press, 1993.
## Related category• NUMBER THEORY | |||||

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