The Diophantine Equation x n −Ny n = z

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

On the Diophantine Equation x^6+ky^3=z^6+kw^3

Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is n...

ON THE DIOPHANTINE EQUATION G n ( x ) = G m ( P ( x ) ) : HIGHER - ORDER RECURRENCES CLEMENS

Let K be a field of characteristic 0 and let (Gn(x))n=0 be a linear recurring sequence of degree d in K[x] defined by the initial terms G0, . . . , Gd−1 ∈ K[x] and by the difference equation Gn+d(x) = Ad−1(x)Gn+d−1(x) + · · ·+A0(x)Gn(x), for n ≥ 0, with A0, . . . , Ad−1 ∈ K[x]. Finally, let P (x) be an element of K[x]. In this paper we are giving fairly general conditions depending only on G0, ...

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....

Thue ’ s theorem and the diophantine equation x 2 − Dy 2 = ± N KEITH MATTHEWS

A constructive version of a theorem of Thue is used to provide representations of certain integers as x2 −Dy2, where D = 2, 3, 5, 6, 7.