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On perfect powers that are sums of cubes of a seven term arithmetic progression. (English) Zbl 07207003
Summary: 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 $$1 \leq r \leq 10^6$$ and $$p \geq 5$$ prime. This article complements the work on the equations $$( x - r )^3 + x^3 + ( x + r )^3 = y^p$$ in [2] and $$( x - 2 r )^3 + ( x - r )^3 + x^3 + ( x + r )^3 + ( x + 2 r )^3 = y^p$$ in [1]. The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.
MSC:
 11D61 Exponential Diophantine equations 11D41 Higher degree equations; Fermat’s equation 11D59 Thue-Mahler equations 11J86 Linear forms in logarithms; Baker’s method
Software:
Magma; PARI/GP; SageMath
Full Text:
References:
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