On a cubic family of Thue equations involving Fibonacci numbers and powers of two

Abstract

In this paper we completely solve the family of parametrised Thue equations

where F_n is the n-th Fibonacci number. In particular, for any integer n ≥ 3 the Thue equation has only the trivial solutions (±1; 0); (0;∓1);∓(F_n; 1);∓(2ⁿ; 1).

Mathematics Subject Classification (2020):

• 11D59
• 11D25
• 11B37

Key words:

• Thue equations
• Thomas' conjecture
• recurrence sequences