Linear combinations of prime powers in X-coordinates of Pell equations

Abstract

Let $$\{X_\ell \}_{\ell \ge 1}$$ be the sequence of X-coordinates of the positive integer solutions (X, Y) of the Pell equation $$X^2-dY^2=\pm 1$$ corresponding to a nonsquare integer $$d>1$$ . We show that there are only a finite number of nonsquare integers $$d > 1$$ such that there are at least two different elements of the sequence $$\{X_\ell \}_{\ell \ge 1}$$ that can be represented as a linear combination of prime powers with fixed primes and coefficients, restricted to the condition that the exponent of the largest prime is the greatest of all exponents. Moreover, we solve explicitly the case in which two of the X-coordinates above are a sum of a power of two and a power of three under the above condition on the exponents. This work is motivated by the recent paper Bertok et al. (Int J Number Theory 13(02):261–271, 2017).