On perfect powers that are sums of cubes of a nine term arithmetic progression. (English) Zbl 07869080
Summary: We study the equation \(( x - 4 r )^3 + ( x - 3 r )^3 + ( x - 2 r )^3 + ( x - r )^3 + x^3 + ( x + r )^3 + ( x + 2 r )^3 + ( x + 3 r )^3 + ( x + 4 r )^3 = y^p\), which is a natural continuation of previous works carried out by A. Argáez-García and the fourth author (perfect powers that are sums of cubes of a three, five and seven term arithmetic progression). Under the assumptions \(0 < r \leq 1 0^6\), \(p \geq 5\) a prime and \(\gcd ( x , r ) = 1\), we show that solutions must satisfy \(x y = 0\). Moreover, we study the equation for prime exponents 2 and 3 in greater detail. Under the assumptions \(r > 0\) a positive integer and \(\gcd ( x , r ) = 1\) we show that there are infinitely many solutions for \(p = 2\) and \(p = 3\) via explicit constructions using integral points on elliptic curves. We use an amalgamation of methods in computational and algebraic number theory to overcome the increased computational challenge. Most notable is a significant computational efficiency obtained through appealing to Bilu, Hanrot and Voutier’s Primitive Divisor Theorem and the method of Chabauty, as well as employing a Thue equation solver earlier on.
MSC:
| 11D61 | Exponential Diophantine equations |
| 11D41 | Higher degree equations; Fermat’s equation |
| 11D25 | Cubic and quartic Diophantine equations |
| 14G05 | Rational points |
Software:
MagmaReferences:
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