51. On perfect powers that are sums of cubes of a nine term arithmetic progression
- Author
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Coppola, Nirvana, Curcó-Iranzo, Mar, Khawaja, Maleeha, Patel, Vandita, and Ülkem, Özge
- Subjects
Mathematics - Number Theory ,Primary 11D61, Secondary 11D41, 11D59, 11J86, 14H52 - Abstract
We study the equation $(x-4r)^3 + (x-3r)^3 + (x-2r)^3+(x-r)^3 + x^3 + (x+r)^3+(x+2r)^3 + (x+3r)^3 + (x+4r)^3 = y^p$, which is a natural continuation of previous works carried out by A. Arg\'{a}ez-Garc\'{i}a and the fourth author (perfect powers that are sums of cubes of a three, five and seven term arithmetic progression). Under the assumptions $0 < r \leq 10^6$, $p \geq 5 $ a prime and $\gcd(x, r) = 1$, we show that solutions must satisfy $xy=0$. 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., Comment: 12 pages
- Published
- 2023