On the Diophantine equation $2^{a} X^{4} + 2^{b} Y^{4} = 2^{c} Z^{4}$

The Diophantine Equation x4 + 2 y4 = z4 + 4 w4

The Diophantine Equation x4 ± y4 = iz2 in Gaussian Integers

The Diophantine equation x4 ± y4 = z2, where x, y and z are integers was studied by Fermat, who proved that there exist no nontrivial solutions. Fermat proved this using the infinite descent method, proving that if a solution can be found, then there exists a smaller solution (see for example [1], Proposition 6.5.3). This was the first particular case proven of Fermat’s Last Theorem (which was ...

On the Diophantine Equation

In this paper, we study the Diophantine equation x2 + C = 2yn in positive integers x, y with gcd(x, y) = 1, where n ≥ 3 and C is a positive integer. If C ≡ 1 (mod 4) we give a very sharp bound for prime values of the exponent n; our main tool here is the result on existence of primitive divisors in Lehmer sequence due Bilu, Hanrot and Voutier. When C 6≡ 1 (mod 4) we explain how the equation can...

On the Diophantine Equation

= c for some integers a, b, c with ab 6= 0, has only finitely many integer solutions. Stoll & Tichy proved more generally that if a, b, c ∈ Q and ab 6= 0, then for m > n ≥ 3, the above equation has only finitely many integral solutions x, y. Independently, Rakaczki established a more precise finiteness result on this binomial equation and extended this result to more general equations (see Acta...

On the Diophantine Equation

If a, b and n are positive integers with b ≥ a and n ≥ 3, then the equation of the title possesses at most one solution in positive integers x and y, with the possible exceptions of (a, b, n) satisfying b = a + 1, 2 ≤ a ≤ min{0.3n, 83} and 17 ≤ n ≤ 347. The proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometri...