This paper is devoted to the generalized Fermat equation xp + yq = zr, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infinite families of generalized Fermat equations that appear amenable to current techniques. While the main tools we employ are based upon the modularity of Galois representations (as is indeed true with all previously solved infinite families), in a number of cases we are led via descent to appeal to a rather intricate combination of multi-Frey techniques.

Generalized Fermat equations: A miscellany

Suppose that $x$ is odd, $n\geq7$ and $p\notin\{2,5\}$ are primes. In this paper, we prove that the Diophantine equations $x^{2}\pm5^{\alpha}p^{n}=y^{n}$ have no solutions in positive integers $\alpha,x,y$ with $gcd(x,y)=1$.

A note on the Diophantine equations $x^{2}\pm5^{\alpha}\cdot p^{n}=y^{n}$

pdf/differences-between-perfect-powers-prime-power-gaps-1fusaz5hmj.pdf

Differences between perfect powers: prime power gaps

We generalize the main result of the paper by Bennett and Mulholland [‘On the diophantine equation x n + y n =2 α pz 2 ’, C. R. Math. Acad. Sci. Soc. R. Can. 28 (2006), 6–11] concerning the solubility of the diophantine equation x n + y n =2 α pz 2 . We also demonstrate, by way of examples, that questions about solubility of a class of diophantine equations of type (3,3, p ) or (4,2, p ) can be reduced, in certain cases, to studying several equations of the type ( p , p ,2).

/pdf/on-a-class-of-generalized-fermat-equations-54ah97clfl.pdf

On a class of generalized fermat equations

In this thesis we study the Diophantine equation
xp - Dy2p = z2; gcd(x; z) = 1; p prime:
We combine two approaches:
- The modular approach using in Wiles's proof of Fermat's Last Theorem.
- Elementary quadratic reciprocity.
We show how using this combination of approaches and computer calculations we can get congruence conditions for the exponent p.

Modular and reciprocity approaches to a family of diophantine equations

Let be a prime number ≥ 5 and be non zero natural numbers. Using the works of K. Ribet and A. Wiles on the modular representations, we get new results about the description of the primitive solutions of the diophantine equation , in case the product of the prime divisors of divides , with an odd prime number. For instance, under some conditions on , we provide a constant such that there are no such solutions if . In application, we obtain information concerning the -rational points of hyperelliptic curves given by the equation with .

Quelques resultats sur les equations ax^p + by^p = cz^2

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Quelques resultats sur les equations ax^p + by^p = cz^2