Original scientific paper
On the power values of the sum of three squares in arithmetic progression
Maohua Le
; Institute of Mathematics, Lingnan Normal College, Zhangjiang, Guangdong, PR China
Gokhan Soydan
; Department of Mathematics, Bursa Uludag University, Bursa, Turkey
Abstract
In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ (*) when $n$ is an odd prime and $d=p^r$, $p>3$ a prime. So this improves the results on the papers of A. Koutsianas and V. Patel \cite{KP} and A. Koutsianas \cite{Kou}. Secondly, under the assumption of our first result, we prove that (*) has at most one solution $(x,y)$. Next, for a general $d$, we prove the following two results: (i) if every odd prime divisor $q$ of $d$ satisfies $q\not\equiv \pm 1 \pmod{2n},$ then (*) has only the solution $(x,y,d,n)=(21,11,2,3)$. (ii) if $n>228000$ and $d>8\sqrt{2}$, then all solutions $(x,y)$ of (*) satisfy $y^n<2^{3/2}d^3$.
Keywords
polynomial Diophantine equation, power sums, primitive divisors of Lehmer sequences, Baker's method
Hrčak ID:
284806
URI
Publication date:
13.11.2022.
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