Diophantus and the diophantine equations By Marta Fuentes Moreno

BIOGRAPHY Diophantus of Alexandria born probably between AD 201 and 215; died aged 84, probably sometime between AD 285 and 299), sometimes called "the father of algebra", was an Alexandrian Greek mathematician and the author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations. He was one of the first mathematicians to introduce symbolism into algebra .

DIOPHANTINE EQUATIONS In mathematics , a Diophantine equation is a polynomial equation , usually in two or more unknowns , such that only the integer solutions are sought or studied (an integer solution is a solution such that all the unknowns take integer values).

AN EQUATION This Diophantine equation has a solution (where x and y are integers) if and only ifcis a multiple of the greatest common divisor ofaandb. Moreover, if(x, y)is a solution, then the other solutions have the form(x + kv, y − ku), wherekis an arbitrary integer, anduandvare the quotients ofaandb(respectively) by the greatest common divisor ofaandb.

PROOF: If d is the greatest common divisor, the identity of Bézout affirms the existence of integers e and f such that e + bf = d. If c is a multiple of d, then c = dh for some integer h, and (hey, fh) it is a solution. Moreover, for each pair of integers x and y, the greatest common divisor d divides a and b ax + by. Therefore, if the equation has a solution, then c must be a multiple of d. If a = b = ud and vd, then for each solution (x, y), we have (x + kv) + b (Y - ku) = ax + by + k (AV - bu) = ax + by + k (UDV - VDU) = ax + by, showing that (x + kv, and - ku) is another solution

EXAMPLES ax + by = 1 This is a linear equation Diophantine. w 3 + x 3 = y 3 + z 3Fermat in 1637 and proved by Wiles in 1995 [3]) states that there are no positive integer solutions (x, y, z The smallest nontrivial solution of the positive integers is 12 3 + 1 3 = 9 3 + 10 3 = 1729. It was given as a clear property famous 1729, a number of taxis (also called Hardy-Ramanujan number) by Ramanujan that Hardy to join in 1917. [1] There are an infinite number of non-trivial solutions. x n + y n = z n For n = 2, there are an infinite number of solutions (x, y, z): the Pythagorean triples. In larger integers n, Fermat's Last Theorem (initially claimed by).

x 2 - ny 2 = ± 1 This is the equation of Pell, which is named after the English mathematician John Pell. It was studied by Brahmagupta in the 7th century, and by Fermat in the 17th century. 4/n = 1/x + 1/y + 1/z The conjecture of Erdos-Strauss points out that, for each positive integer n ≥ 2, there is a solution in x, y, and z, the numbers positive integers. Although not usually shown as a polynomial, this example is equivalent to the polynomial equation 4 yzn + xyz = x and n = n AVN + (xy + xz + yz). x 4 + y 4 + z 4 = w 4 Incorrectly conjectured by Euler not have nontrivial solutions. Tested by Elkies have an infinite number of nontrivial solutions with a computer search Frye determine the smallest nontrivial solution