# On perfect powers that are sums of cubes of a five term arithmetic progression

@article{ArgaezGarcia2019OnPP, title={On perfect powers that are sums of cubes of a five term arithmetic progression}, author={Alejandro Arg'aez-Garc'ia}, journal={Journal of Number Theory}, year={2019} }

Using only elementary arguments, Cassels and Uchiyama (independently) determined all squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result and determined all perfect powers that are sums of three consecutive cubes. Recently, the equation $(x-r)^k + x^k + (x+r)^k$ has been studied for $k=4$ by Zhongfeng Zhang and for $k=2$ by Koutsianas. In this paper, we complement the work of Cassels, Koutsianas and Zhang by considering the case when $k=3$ and showing that the…

#### 10 Citations

Perfect powers in sum of three fifth powers

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In this paper, we determine all primitive solutions to the equation $(x+r)^2 +(x+2r)^2 +\cdots +(x+dr)^2 = y^n$ for $2\leq d\leq 10$ and for $1\leq r\leq 10^4$. We make use of a factorization…

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Abstract We prove that the equation ( x − 3 r ) 3 + ( x − 2 r ) 3 + ( x − r ) 3 + x 3 + ( x + r ) 3 + ( x + 2 r ) 3 + ( x + 3 r ) 3 = y p only has solutions which satisfy x y = 0 for 1 ≤ r ≤ 10 6 and…

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In this paper, we solve the equation of the title under the assumption that $\gcd(x,d)=1$ and $n\geq 2$. This generalizes earlier work of the first author, Patel and Siksek [BPS16]. Our main tools…

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We determine primitive solutions to the equation [Formula: see text] for [Formula: see text], making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.

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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, firstly, we give an explicit formula for all positive integer…

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We prove that the equation ${ (x - y)^4 + x^4 + (x + y)^4 = z^n }$ has no integer solutions ${ x, y, z}$ with ${ \gcd(x, y) = 1 }$ for all integers ${ n > 1 }$. We mainly use a modular approach with…

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In this paper, we determine all primitive solutions to the equation $(x+r)^2 +(x+2r)^2 +\cdots +(x+dr)^2 = y^n$ for $2\leq d\leq 10$ and for $1\leq r\leq 10^4$. We make use of a factorization…

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Abstract We prove that the equation ( x − 3 r ) 3 + ( x − 2 r ) 3 + ( x − r ) 3 + x 3 + ( x + r ) 3 + ( x + 2 r ) 3 + ( x + 3 r ) 3 = y p only has solutions which satisfy x y = 0 for 1 ≤ r ≤ 10 6 and…

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In this paper, we determine the primitive solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ when $n\geq 2$ and $d=p^b$, $p$ a prime and $p\leq 10^4$. The main ingredients are the…

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