An extension of a theorem of Euler

Rational function variant of a problem of Erdös and Graham

“…Bennett, Bruin, Győry and Hajdu ( [2]) solved (1.1) when 6 ≤ k ≤ 11. Hirata-Kohno, Laishram, Shorey and Tijdeman ( [13]) completely solved the equation (1.1) with 3 ≤ k < 110. Combining their result with those of Tengely ( [23]) all solutions of (1.1) with 3 ≤ k ≤ 100, P (b) < k are determined, where P (u) denotes the greatest prime factor of u, with the convention P (1) = 1.…”

On a generalization of a problem of Erdos and Graham

“…Bennett, Bruin, Győry and Hajdu [3] solved (1) with 6 ≤ k ≤ 11 and l = 2. Hirata-Kohno, Laishram, Shorey and Tijdeman [22] completely solved (1) with 3 ≤ k < 110. Now assume for this paragraph that l ≥ 3.…”

Variations on a theme of Runge: effective determination of integral points on certain varieties

“…When k = 5, Obláth [18] proved that (5.5) does not hold if b = 1 and Mukhopadhyay and Shorey [15] handled the general case P (b) < k. When 6 ≤ k ≤ 11, P (b) ≤ 5, Bennett, Bruin, Győry, and Hajdu [2] showed that the only solution to (5.5) is k = 6, d = 1, x = 1. When 8 ≤ k ≤ 100, d > 1, Hirata-Kohno, Laishram, Shorey, and Tijdeman [9] showed that (5.5) does not hold except possibly in a small number of exceptional cases. These remaining exceptional cases were handled by Tengely [28].…”