On certain sequences in Mordell–Weil type groups

In this paper we investigate divisibility properties of two families of sequences in the Mordell–Weil group of elliptic curves over number fields without complex multiplication. We also consider more general groups of Mordell–Weil type. M. Ward ([W], Theorem 1.) proved that a linear integral recurring sequence of order two which is not nontrivially degenerate has an infinite number of distinct prime divisors, where by a divisor of a sequence we mean a positive integer dividing some term of the sequence. Then L. Somer ([Som]) using a result by A. Schinzel ([Schi2]) determined those recurrences that have almost all primes as divisors. The general terms of nondegenerate linear recurring sequences of order two are of the form αA− βB and the general terms of trivially degenerate linear recurring sequences of order two are of the form α(A+ nB). In the present paper we investigate analogues of such sequences in Mordell–Weil group of elliptic curves: Let F be a number field, E/F an elliptic curve without complex multiplication, P,Q ∈ E(F ) and φ, ψ be isogenies (since we deal with curves without CM the isogenies are simply endomorphisms defined by the multiplication by rational integers; see Remark 5). We investigate sequences: