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Diophantine equation

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.

Examples of Diophantine equations

In the following Diophantine equations, x, y, and z are the unknowns, the other letters being given.

$ax+by=1,$

This is a linear Diophantine equation (see the section "Linear Diophantine equations" below).

$x^n+y^n=z^n,$

For n = 2 there are infinitely 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 equation exist.

$x^2-ny^2=pm 1,$

(Pell's equation) which is named after the English mathematician John Pell. It was studied by Brahmagupta in the 7th century, as well as by Fermat in the 17th century.

$frac\left\{4\right\}\left\{n\right\} = frac\left\{1\right\}\left\{x\right\} + frac\left\{1\right\}\left\{y\right\} + frac\left\{1\right\}\left\{z\right\}$

The Erdős–Straus conjecture states that, for every positive integer n ≥ 2, there exists a solution with x, y, and z all positive integers.

Diophantine analysis

Typical questions

The questions asked in Diophantine analysis include:

1. Are there any solutions?
2. Are there any solutions beyond some that are easily found by inspection?
3. Are there finitely or infinitely many solutions?
4. Can all solutions be found, in theory?
5. 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.

Diophantine analysis in India

India's contribution to integral solutions of Diophantine equations can be traced back to the Sulba Sutras, which were Indian mathematical texts written between 800 BC and 500 BC. Baudhayana (circa 800 BC) finds two sets of positive integral solutions to a set of simultaneous Diophantine equations, and also attempts simultaneous Diophantine equations with up to four unknowns. Apastamba (circa 600 BC) attempts simultaneous Diophantine equations with up to five unknowns.

Diophantine equations were later extensively studied by mathematicians in medieval India, who were the first to systematically investigate methods for determination of integral solutions of Diophantine equations. Systematic methods for finding integer solutions of Diophantine equations could be found in Indian texts from the time of Aryabhata AD (499). The first explicit description of the general integral solution of the linear Diophantine equation ay + bx = c occurs in his text Aryabhatiya. This algorithm is considered to be one of the most significant contributions of Aryabhata in pure mathematics. The technique was applied by Aryabhata to give integral solutions of simultaneous Diophantine equations of first degree, a problem with important applications in astronomy.

Aryabhata describes the algorithm in just two stanzas of Aryabhatiya. His cryptic verses were elaborated by Bhaskara I (6th century) in his commentary Aryabhatiya Bhasya. Bhaskara I illustrated Aryabhata's rule with several examples including 24 concrete problems from astronomy. Without the explanation of Bhaskara I, it would have been difficult to interpret Aryabhata's verses. Bhaskara I aptly called the method kuttaka (pulverisation). The idea in kuttaka was later considered so important by the Indians that initially the whole subject of algebra used to be called kuttaka-ganita, or simply kuttaka.

Brahmagupta (628) handled more difficult Diophantine equations - he investigated Pell's equation, and in his Samasabhavana he laid out a procedure to solve Diophantine equations of the second order, such as 61x2 + 1 = y2. These methods were unknown in the west, and this very equation was posed as a problem in 1657 by the French mathematician Pierre de Fermat; its solution was found seventy years later by Euler. However, the solution to this equation had been recorded centuries earlier by Bhaskara II (1150) (who also found the solution to Pell's equation), using a modified version of Brahmagupta's method.

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 an + bn = cn 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.

In 1657, Fermat attempted the Diophantine equation 61x2 + 1 = y2 (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.

Hilbert's tenth problem

In 1900, in recognition of their depth, David 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 techniques from algebraic geometry in this field, has continued to grow as a result; since treating arbitrary equations is a dead end, attention turns to 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 system of simultaneous equations, which is a vector in a prescribed field K, when K is not algebraically closed.

Modern research

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 equivalently described as recursively enumerable. In other words, the general problem of Diophantine analysis is blessed or cursed with universality, and in any case is not something that will be solved except by re-expressing it in other terms.

The field of Diophantine approximation deals with the cases of Diophantine inequalities. Here 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.

The most celebrated single question in the field, the conjecture known as Fermat's Last Theorem, was cleared up by Andrew Wiles. His journey through this proof can be found here: Other major results, such as Faltings' theorem, have disposed of old conjectures.

Linear Diophantine equations

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has infinitely many solutions. These can be found by applying the extended Euclidean algorithm. It follows that there are also infinitely many solutions if c is a multiple of the greatest common divisor of a and b. If c is not a multiple of the greatest common divisor of a and b, then the Diophantine equation ax + by = c has no solutions.

Exponential Diophantine equations

If a Diophantine equation has as an additional variable or variables occurring as exponents, it is an exponential Diophantine equation. One example is the Ramanujan-Nagell equation, 2n − 7 = x2. Such equations do not have a general theory; particular cases such as Catalan's conjecture have been tackled, however the majority are solved via ad hoc methods such as Størmer's theorem or even trial and error.