On Pillai's problem with X-coordinates of Pell equations and powers of 2

Lucas Numbers Which Are Concatenations of Two Repdigits

“…Lemma 3. (see [24]) Let τ be an irrational number, M be a positive integer and p k q k (k = 0, 1, 2, . .…”

“…For k = 2 and k = 3, the Diophantine Equation ( 23) has been solved in [18] and [21], respectively. In this paper, we solve the Diophantine Equation (24) for the case of k = 2. Our future research work is to solve the Diophantine Equations ( 23) and ( 24) completely for the case of k ≥ 3.…”

On Y-coordinates of Pell equations which are base 2 rep-digits

“…Over the 27401 such pairs we obtained only 55 vectors. They all had Y 1 = 1, k 2 = 3 and d = 2 2ℓ − 1, for ℓ ∈ [5,59]. Thus, these are exactly the parametric family mentioned after the statement of the main theorem which are in the tested range so by the argument from the end of the proof of Lemma 4.1 there cannot be a third solution (k 3 , n 3 ).…”

“…Since k 3 ≤ n 3 , we get k 3 ≤ 255. Now we generated, for each of the 55 values of d, the numbers Y k for k odd in [5,255] and tested whether Y k + 1 is a power of 2. No additional example was found.…”

2-Peble Triangles Over Figurate Numbers

“…The articles that use Lucas numbers (1) , x-coordinate of Pell's equation with powers of 2 (2) , and Fibonacci numbers (3) in Pell equations served as inspirations for this article. New types of rectangles and triangles are defined and some of their properties are explored by using Diophantine equations in articles (4,5) .…”