On the diophantine equation x2 + p2k = yn

Abstract

Let 2 ≤ p < 100 be a rational prime and consider equation (3) in the title in integer unknowns x, y, n, k with x > 0, y > 1, n ≥ 3 prime, k ≥ 0 and gcd(x, y) = 1. Under the above conditions we give all solutions of the title equation (see the Theorem). We note that if in (3) gcd(x, y) = 1, our Theorem is an extension of several earlier results [15], [

On the diophantine equation x 2 + 2a · 3b · 11c = y n

“…So ±3 β 1 11 γ 1 ≡ 5 (mod 8), showing that the sign on the left hand side is negative and β 1 + γ 1 is odd.…”

“…Partial results on the case when S = {7} appear in [23]. Recently, Bérczes and Pink [8], found all the solutions of the Diophantine equation (1.1) when C = p k and k is even, where p is any prime in the interval [2,100].…”

On the Diophantine equation 1+2a+xb=yn

“…In the course of the proof of our main result we have to solve completely several equations of the shape (6) x b + t = y n , where x, t are given odd positive integers and b, y, n are unknown positive integers with y ≥ 2 even and n ≥ 3 an odd prime. By using Lemma 3.2 and Lemma 3.1 we derive relatively sharp explicit upper bounds for n in equation (6).…”

“…So, in what follows, we may write b occurring in (6) in the form b = nB + r, where B and r are integers for which B ≥ 0 and 0 < r ≤ n − 1. Thus, equation (6) yields…”

“…, a s are unknown positive integers. For related results we only refer to the papers of Schinzel and Tijdeman [21], Cohn [12], Bugeaud, Mignotte and Siksek [10], Luca [18], Bugeaud and Muriefah [11], Bérczes and Pink [6], Le and Zhu [16] and Xiaowei [23], and the references given there.…”