On common terms of linear recurrences

Polynomial-exponential equations and linear recurrences

“…Then assuming x, q > 1 the solutions q of (1.6) can be bounded by an effectively computable constant which depends on E and the coefficients and initial values of the recurrence. Kiss [11] proved that, in fact, q is less than a number which is effectively computable in terms of the greatest prime divisor of E and the coefficients and the initial values of the sequence (G n ).…”

“…For second order recurrences (G n ) with |A 2 | = 1 Nemes and Pethő [13] characterized all polynomials P for which the equation G n = P (x) has infinitely many solutions (see also [16]). Kiss [11] and Shorey and Stewart [25] dealt with equation (1.8) for nondegenerate linear recurring sequences (G n ) of arbitrary order, under condition (1.9) and the additional assumptions that d is the degree of α 1 over Q, α 1 and α 2 are multiplicatively independent and α 2 = ±1. Then they showed that there are only finitely many integers n, x and q with n ≥ 0, |x| > 1 and…”

Perfect powers in linear recurring sequences

“…Then assuming x, q > 1 the solutions q of (3) can be bounded by an effectively computable constant which depends on the coefficients and initial values of the recurrence. Kiss [8] proved that, in fact, q is less than a number which is effectively computable in terms of the greatest prime divisor of E and the coefficients and the initial values of the sequence (G n ).…”

“…Nemes and Pethő were also able to show that if q is a fixed integer larger than one and (6) has infinitely many integral solutions n and x, then T (x) can be characterized in terms of the Chebyshev polynomials. Kiss [8] and Shorey and Stewart [20] dealt with equation (6) for nondegenerate linear recurring sequences (G n ) of arbitrary order, under condition (7) and the additional assumptions that E = 1 and d is the degree of α 1 over Q, α 1 and α 2 are multiplicatively independent and α 2 = ±1. They showed that there are then only finitely many integers n, x and q with n ≥ 0, |x| > 1 and…”

Effective solution of two simultaneous Pell equations by the elliptic logarithm method

“…A third category includes papers in which effective but not explicit results are proved; see [19], [22], [23].…”