A note on the Diophantine equation \((x+1)^3 + (x+2)^3 + \cdots + (2x)^3 = y^n\). (English) Zbl 1510.11090
For fixed positive integers \(k,l,n\), the Diophantine equation
\[
(x+1)^k+(x+2)^k+\cdots + (lx)^k = y^n
\]
has attracted the interest of various authors. In this paper the author considers the case when \(l=2,k=3\) and \(n\ge 2\) which results to the title equation. By a clever use of totally elementary Number Theory, he proves that this equation has no integer solutions with \(x,y\ge 1\).
Reviewer: Nikos Tzanakis (Iraklion)
MSC:
| 11D41 | Higher degree equations; Fermat’s equation |
| 11A05 | Multiplicative structure; Euclidean algorithm; greatest common divisors |
| 11A07 | Congruences; primitive roots; residue systems |
| 11D61 | Exponential Diophantine equations |
References:
| [1] | M. Bai and Z. Zhang, On the Diophantine equation .x C 1/ 2 C .x C 2/ 2 C C .x C d / 2 D y n , Funct. |
| [2] | Approx. Comment. Math. 49 (2013), 73-77. · Zbl 1335.11023 |
| [3] | D. Bartoli and G. Soydan, The Diophantine equation .x C 1/ k C .x C 2/ k C C .lx/ k D y n revisited, Publ. Math. Debrecen 96 (2020), no. 1-2, httpsW//arxiv.org/abs/1909.06100v1. |
| [4] | A. Bérczes, I. Pink, G. Savas and G. Soydan, On the equation .x C 1/ k C .x C 2/ k C C .2x/ k D y n , J. Number Theory, 183 (2017), 326-351. · Zbl 1433.11034 |
| [5] | G. Soydan, On the Diophantine equation .x C 1/ k C .x C 2/ k C C .lx/ k D y n , Publ. Math. Debrecen 91 (2017), no. 3-4, 369-382. · Zbl 1413.11074 |
| [6] | Dai Co Viet Road, Ha Noi, Vietnam tho.nguyenxuan1@hust.edu.vn |
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