Shifted powers in Lucas–Lehmer sequences

Shifted powers in binary recurrence sequences

Let {uk} be a Lucas sequence. A standard technique for determining the perfect powers in the sequence {uk} combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation uk = xn can be translated into a ternary equation of the form ay2 = bx2n + c (with a, b, c ∈ Z) for which Frey curve...

On shifted products which are powers

Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj + 1 is a square for 1 ≤ i < j ≤ 4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grin...

Shifted products that are coprime pure powers

A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab + 1 of two different elements a and b of A is always a pure power, and the occurring pure powers are all coprime. We prove that each coprime Diophantine powerset A ⊂ {1, . . . , N} has |A| 8000 log N/ log log N for sufficiently large N. The proof combines results from extremal graph theory with numbe...

On the linear independence of shifted powers

We call shifted power a polynomial of the form (x− a)e. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family F of shifted powers are linearly independent or, failing that, to give a lower bound on the dimension of the space of polynomials spanned by F . In particular, we give simple criteria ensuring that the dimension of the span of...

Powers in Sturmian sequences