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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

Puni tekst: engleski pdf 136 Kb

str. 137-150

preuzimanja: 171



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 $y^n<2^{3/2}d^3$.

Ključne riječi

polynomial Diophantine equation, power sums, primitive divisors of Lehmer sequences, Baker's method

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Posjeta: 393 *