a diophantine equation?

Question

Any number having any power (>1)can be expressed as difference of two perfect coprime squares.
can this be used successfully to prove that the diophantine equation
x^(m) = a^(n) - b^(n)
has no coprime solutions for x,a & b when m>2 & n>2.
here m =/= n.

Answer 1

This would be a special case of the Fermat-Catalan Conjecture.

This conjecture is that there are only a finite number of solutions of x^p + y^q = z^r for 1/p + 1/q + 1/r < 1. Ten solutions are known, but all of them have the three exponents different (except for 1^p + 2^3 = 3^2) and one of the exponents equal to 2.

Your equation would be the special case that there are no solutions with p > 2 and q = r. It is already known that there are no solutions with p = q and r > 2 (Darmon and Merel, 1997). Any proof that there was another special case with no solutions would be of great interest, but its proof seems unlikely to be as elementary as you hope.

Answer 2

In mathematics, a Diophantine equation is an indeterminate polynomial equation that only allows the variables to be integers. 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 CE, Diophantus of Alexandria, Egypt 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 at many times, the formulation of general theories of Diophantine equations (further to the theory of quadratic forms) was an achievement of the twentieth century.

In the following, 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).
xn + yn = zn: 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 above equation exist.
x2 - n y2 = 1: (Pell's equation) which is named, mistakenly, after the English mathematician John Pell. It was studied by Brahmagupta in the 6th century and much later by Fermat.
, where and : These are the Thue equations, and are, in general, solvable.
4/n = 1/x + 1/y + 1/z, or, in polynomial form, 4xyz=n(xy+xz+yz). The Erdős–Straus conjecture states that, for every positive integer n, there exists a solution with x, y, and z all positive integers