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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
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References:
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