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 are constants.
$ax+by=1\backslash ,$	This is a linear Diophantine equation (see the section "Linear Diophantine equations" below).
$x^n+y^n=z^n\; \backslash ,$	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.
$x^2-ny^2=\backslash pm\; 1\backslash ,$	(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.
$\backslash frac\{4\}\{n\}\; =\; \backslash frac\{1\}\{x\}\; +\; \backslash frac\{1\}\{y\}\; +\; \backslash frac\{1\}\{z\}$	The Erdős–Straus conjecture states that, for every positive integer n ≥ 2, there exists a solution with x, y, and z all positive integers. Although not usually stated in polynomial form, this example is equivalent to the polynomial equation 4xyz = yzn + xzn + xyn = n(yz + xz + xy).

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 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.

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 solved by Andrew Wiles but using tools from algebraic geometry developed during the last century rather than within number theory where the conjecture was originally formulated. 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 an infinite number of solutions. These can be found by applying the extended Euclidean algorithm. It follows that there are also infinite 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.

Notes

References

External links

Diophantine Equation. From MathWorld at Wolfram Research.

Diophantine Equation. From PlanetMath.

Dario Alpern's Online Calculator. Retrieved 18 March 2009

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Name	Pierre de Fermat
Caption	Pierre de Fermat
Birth place	Beaumont-de-Lomagne, France
Death date	January 12, 1665
Death place	Castres, France
Residence	France
Nationality	French
Field	Mathematics and Law
Known for	Analytic geometry Fermat's principle Probability Fermat's Last Theorem}}

Pierre de Fermat (; 17 August 1601 or 1607/8 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of the then unknown differential calculus, as well as his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for Fermat's Last Theorem, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

Life and work

Fermat was born in Beaumont-de-Lomagne, Tarn-et-Garonne, France; the late 15th century mansion where Fermat was born is now a museum. He was of Basque origin. Fermat's father was a wealthy leather merchant and second consul of Beaumont-de-Lomagne. Pierre had a brother and two sisters and was almost certainly brought up in the town of his birth. There is little evidence concerning his school education, but it may have been at the local Franciscan monastery.

He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius's De Locis Planis to one of the mathematicians there. Certainly in Bordeaux he was in contact with Beaugrand and during this time he produced important work on maxima and minima which he gave to Étienne d'Espagnet who clearly shared mathematical interests with Fermat. There he became much influenced by the work of Franciscus Vieta.

From Bordeaux, Fermat went to Orléans where he studied law at the University. He received a degree in civil law before, in 1631, receiving the title of councillor at the High Court of Judicature in Toulouse, which he held for the rest of his life. Due to the office he now held he became entitled to change his name from Pierre Fermat to Pierre de Fermat. Fluent in Latin, Basque, classical Greek, Italian, and Spanish, Fermat was praised for his written verse in several languages, and his advice was eagerly sought regarding the emendation of Greek texts.

He communicated most of his work in letters to friends, often with little or no proof of his theorems. This allowed him to preserve his status as an "amateur" while gaining the recognition he desired. This naturally led to priority disputes with fellow contemporaries such as Descartes and Wallis. He developed a close relationship with Blaise Pascal.

Anders Hald writes that, "The basis of Fermat's mathematics was the classical Greek treatises combined with Vieta's new algebraic methods."

Work

Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie. This manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").

In Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation. In these works, Fermat obtained a technique for finding the centers of gravity of various plane and solid figures, which led to his further work in quadrature. Fermat was the first person known to have evaluated the integral of general power functions. Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series. The resulting formula was helpful to Newton, and then Leibniz, when they independently developed the fundamental theorem of calculus.

In number theory, Fermat studied Pell's equation, perfect numbers, amicable numbers and what would later become Fermat numbers. It was while researching perfect numbers that he discovered the little theorem. He invented a factorization method - Fermat's factorization method - as well as the proof technique of infinite descent, which he used to prove Fermat's Last Theorem for the case n = 4. Fermat developed the two-square theorem, and the polygonal number theorem, which states that each number is a sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on.

Although Fermat claimed to have proved all his arithmetic theorems, few records of his proofs have survived. Many mathematicians, including Gauss, doubted several of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat. His famous Last Theorem was first discovered by his son in the margin on his father's copy of an edition of Diophantus, and included the statement that the margin was too small to include the proof. He had not bothered to inform even Marin Mersenne of it. It was not proved until 1994, using techniques unavailable to Fermat.

Although he carefully studied, and drew inspiration from Diophantus, Fermat began a different tradition. Diophantus was content to find a single solution to his equations, even if it were an undesired fractional one. Fermat was interested only in integer solutions to his Diophantine equations, and he looked for all possible general solutions. He often proved that certain equations had no solution, which usually baffled his contemporaries.

Through his correspondence with Pascal in 1654, Fermat and Pascal helped lay the fundamental groundwork for the theory of probability. From this brief but productive collaboration on the problem of points, they are now regarded as joint founders of probability theory. Fermat is credited with carrying out the first ever rigorous probability calculation. In it, he was asked by a professional gambler why if he bet on rolling at least one six in four throws of a die he won in the long term, whereas betting on throwing at least one double-six in 24 throws of two dice resulted in him losing. Fermat subsequently proved why this was the case mathematically.

Fermat's principle of least time (which he used to derive Snell's law in 1657) was the first variational principle enunciated in physics since Hero of Alexandria described a principle of least distance in the first century CE. In this way, Fermat is recognized as a key figure in the historical development of the fundamental principle of least action in physics. The term Fermat functional was named in recognition of this role.

Death

He died at Castres, Tarn.

Of Fermat's number theoretic work, the great 20th century mathematician André Weil wrote that "... what we possess of his methods for dealing with curves of genus 1 is remarkably coherent; it is still the foundation for the modern theory of such curves. It naturally falls into two parts; the first one ... may conveniently be termed a method of ascent, in contrast with the descent which is rightly regarded as Fermat's own." Regarding Fermat's use of ascent, Weil continued "The novelty consisted in the vastly extended use which Fermat made of it, giving him at least a partial equivalent of what we would obtain by the systematic use of the group theoretical properties of the rational points on a standard cubic." With his gift for number relations and his ability to find proofs for many of his theorems, Fermat essentially created the modern theory of numbers.

See also

Notes

References

Books referenced

Further reading

External links

The Life and times of Pierre de Fermat (1601 - 1665) from W. W. Rouse Ball's History of Mathematics

The Mathematics of Fermat's Last Theorem

Fermat's Achievements

Fermat's Fallibility at MathPages

History of Fermat's Last Theorem (French)

Category:1601 births Category:1665 deaths Category:People from Tarn-et-Garonne Category:17th-century mathematicians Category:17th-century French people Category:Basque mathematicians Category:French mathematicians Category:Number theorists Category:Occitan people Category:French Roman Catholics Category:French lawyers