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In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more general object, and ask about the lattice points on it.
The word Diophantine refers to the Hellenistic mathematician of the 3rd century, 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.
While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the theory of quadratic forms) was an achievement of the twentieth century.
Contents
Examples of Diophantine equations
Diophantine analysis
Typical questions
The questions asked in Diophantine analysis include:
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.
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 a^{n} + b^{n} = c^{n} has no solutions for any n higher than two." And then he wrote, intriguingly: "I have discovered a truly marvelous proof of this, which, however, the margin is not large enough to contain." Such a proof eluded mathematicians for centuries, however. As an unproven conjecture that eluded brilliant mathematicians' attempts to either prove it or disprove it for generations, his statement became famous as Fermat's Last Theorem. It wasn't until 1994 that it was proven by the British mathematician Andrew Wiles.
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