On the Diophantine equation $ax^{2t}+bx^ty+cy^2=d$ and pure powers in recurrence sequences.

Diophantine Properties of Linear Recursive Sequences I

“…For second order recurrences we can bound effectively not only the exponents of the perfect powers appearing in the sequence, but also the largest index for which a term of the sequence can be a perfect power. This was proved by Shorey and Stewart [ShSt1] and independently by me [P1]. Theorem 7.…”

“…There are results only in that case, when G n has a dominating characteristic zero. Under this assumption Shorey and Stewart [ShSt1] proved that q < c 2 provided |y| > 1.…”

“…will denote effectively computable constants depending only on the initial terms and on the coefficients of the characteristic polynomial of G n . For binary recurrences Shorey and Stewart [ShSt1] and independently the author of the present notes [P1] proved that max{n, |y|, q} < c 1 . We come back to this result in section 2.3.…”

On a diophantine equation related to a conjecture of Erdös and Graham

“…For instance, the combined work of Ljunggren [5] and Cohn [2] completely solved the equation T i = x 2 , in which it was shown that equation (2) implies that either i = 1 or i = 2, and that a solution exists for both i = 1, 2 only when d = 1785. More general results on polynomial values in linear recurrence sequences have been proved by Nemes and Pethö [9], and also by Shorey and Stewart [10].…”

Solving parametrized systems of Pell equations

“…We shall need the following theorem (see Shorey & Tijdeman [14], Shorey & Stewart [13], Pethő [9]), which we quote in the special case needed in this paper.…”