On the Diophantine equation $| Ax^n -By^n| = 1$, $n = 5$.

On the Diophantine equation |axn - byn | = 1

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...

On the Diophantine equation q n − 1 q − 1 = y

There exist many results about the Diophantine equation (qn − 1)/(q − 1) = ym, where m ≥ 2 and n ≥ 3. In this paper, we suppose that m = 1, n is an odd integer and q a power of a prime number. Also let y be an integer such that the number of prime divisors of y − 1 is less than or equal to 3. Then we solve completely the Diophantine equation (qn − 1)/(q − 1) = y for infinitely many values of y....

ON THE DIOPHANTINE EQUATION xn − 1 x −

We prove that if (x, y, n, q) 6= (18, 7, 3, 3) is a solution of the Diophantine equation (xn−1)/(x−1) = y with q prime, then there exists a prime number p such that p divides x and q divides p − 1. This allows us to solve completely this Diophantine equation for infinitely many values of x. The proofs require several different methods in diophantine approximation together with some heavy comput...

” THE DIOPHANTINE EQUATION x n = Dy 2 + 1 ”

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...