On the power values of the sum of three squares in arithmetic progression. (English) Zbl 1510.11089
Let \(x, y \in \mathbb{N}\) such that \(gcd(a,b)=1\), \(n \geq 2\) and consider the following Diophantine equation:
\[
(x-d)^2+x^2 + (x+d)^2 = y^n.
\]
and the special case \(d=p^r\) with \(r \in \mathbb{N}\) (which extends the previous results cited in the paper under review).
As a first result of the paper under review, the authors put a criteria on the solutions of the equation above. With this assumption, the authors prove that the equation above has at most one solution \((x, y)\).
Third result is the same result but for general \(d\).
As a final result, they give the necessary conditions for the solutions for large numbers.
The proofs include several techniques such as some tools of algebraic number theory and linear forms in logarithms.
The paper under review is interesting, well-organized and clearly-written.
As a first result of the paper under review, the authors put a criteria on the solutions of the equation above. With this assumption, the authors prove that the equation above has at most one solution \((x, y)\).
Third result is the same result but for general \(d\).
As a final result, they give the necessary conditions for the solutions for large numbers.
The proofs include several techniques such as some tools of algebraic number theory and linear forms in logarithms.
The paper under review is interesting, well-organized and clearly-written.
Reviewer: Ilker Inam (Bilecik)
MSC:
| 11D41 | Higher degree equations; Fermat’s equation |
| 11J86 | Linear forms in logarithms; Baker’s method |
Keywords:
polynomial Diophantine equation; power sums; primitive divisors of Lehmer sequences; Baker’s methodReferences:
| [1] | A. Argáez-Garcia, On perfect powers that are sums of cubes of a five term arithmetic progression, J. Number Theory 201(2019), 460-472. · Zbl 1459.11089 |
| [2] | A. Argáez-Garcia, V. Patel, On perfect powers that are sums of cubes of a three term arithmetic progression, J. Comb. and Number Theory 10(2019), 147-160. · Zbl 1459.11091 |
| [3] | A. Argáez-Garcia, On perfect powers that are sums of cubes of a seven term arith-metic progression, J. Number Theory 214(2020), 440-451. · Zbl 1459.11090 |
| [4] | M. Bai, Z. Zhang, On the Diophantine equation (x+1) 2 +(x+2) 2 +• • •+(x+d) 2 = y n , Functiones Approx. Com. Math. 49(2013), 73-77. · Zbl 1335.11023 |
| [5] | D. Bartoli, G. Soydan,The Diophantine equation (x+1) k +(x+2) k +• • •+(lx) k = y n revisited, Publ. Math. Debrecen 96/1-2(2020), 111-120. · Zbl 1449.11063 |
| [6] | M. A. Bennett, K. Győry,Á. Pintér, On the Diophantine equation 1 k + 2 k + • • • + x k = y n , Compos. Math. 140(2004), 1417-1431. · Zbl 1066.11013 |
| [7] | M. A. Bennett, A. Koutsianas, The equation (x − d) 5 + x 5 + (x + d) 5 = y n , Acta Arith. 198(2021), 387-399. · Zbl 1475.11049 |
| [8] | M. A. Bennett, V. Patel, S. Siksek, Superelliptic equations arising from sums of consecutive powers, Acta Arith. 172(2016), 377-393. · Zbl 1401.11076 |
| [9] | M. A. Bennett, V. Patel, S. Siksek, Perfect powers that are sums of consecutive cubes, Mathematika 63(2016), 230-249. · Zbl 1437.11046 |
| [10] | A. Bérczes, L. Hajdu, T. Miyazaki, I. Pink, On the equation 1 k +2 k +• • •+x k = y n , J. Number Theory 163(2016), 43-60. · Zbl 1402.11052 |
| [11] | A. Bérczes, I. Pink, G. Savaş, G. Soydan, On the Diophantine equation (x + 1) k + (x + 2) k + • • • + (2x) k = y n , J. Number Theory 183(2018), 326-351. · Zbl 1433.11034 |
| [12] | Y. Bilu, G. Hanrot, P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers (with Appendix by Mignotte), J. Reine Angew. Math. 539(2001), 75-122. · Zbl 0995.11010 |
| [13] | J. W. S. Cassels, A Diophantine equation, Glasgow Math. Journal 27(1985), 11-88. · Zbl 0576.10010 |
| [14] | P. Das, P. K. Dey, A. Koutsianas, N. Tzanakis, Perfect powers is sum three fifth powers, arxiv:2008.07804v2 [math.NT], August 28, 2020. |
| [15] | K. Győry,Á. Pintér,On the equation 1 k + 2 k + • • • + x k = y n , Publ. Math. Debrecen 62(2003), 403-414. · Zbl 1026.11042 |
| [16] | L. Hajdu, On a conjecture of Schäffer concerning the equation 1 k + 2 k + • • • + x k = y n , J. Number Theory 155(2015), 129-138. · Zbl 1377.11039 |
| [17] | M. Jacobson,Á.Pintér, G. P. Walsh, A computational approach for solving y 2 = 1 k + 2 k + • • • + x k , Math. Comp. 72(2003), 2099-2110. · Zbl 1113.11075 |
| [18] | A. Koutsianas, On the solutions of the Diophantine equation (x−d) 2 +x 2 +(x+d) 2 = y n for d a prime power, Func. Approx. Comment. Math., 64(2021), 141-151. · Zbl 1505.11060 |
| [19] | A. Koutsianas, V. Patel, Perfect powers that are sums of squares in a three term arithmetic progression, Int. J. Number Theory 14(2018), 2729-2735. · Zbl 1441.11065 |
| [20] | M. -H. Le, Some exponential Diophantine equations I: The equation D1x 2 − D2y 2 = λk z , J. Number Theory, 55(1995), 209-221. · Zbl 0852.11015 |
| [21] | V. A. Lebesgue, Sur l’impossibilité, en nombres entiers, de l’équation x m = y 2 + 1, Nouv. Ann. de Math. 9(1850), 178-181. |
| [22] | D. H. Lehmer, An extended theory of Lucas’ function, Ann. Math. 31(1930), 419-448. · JFM 56.0874.04 |
| [23] | É. Lucas, Question 1180, Nouvelles Ann. Math. 14(1875), 336. |
| [24] | V. Patel, S. Siksek, On powers that are sums of consecutive like powers, Research in Num. Theory (2017), Article 2, 7 pp. · Zbl 1358.11052 |
| [25] | Á. Pintér, A note on the equation 1 k + 2 k + • • • + (x − 1) k = y m , Indag. Math. (N.S.) 8(1997), 119-123. · Zbl 0876.11014 |
| [26] | Á. Pintér, On the power values of power sums, J. Number Theory 125(2007), 412-423. · Zbl 1127.11022 |
| [27] | J. J. Schäffer, The equation 1 p + 2 p + • • • + n p = m q , Acta Math. 95(1956), 155-189. · Zbl 0071.03702 |
| [28] | G. Soydan, On the Diophantine equation (x + 1) k + (x + 2) k + • • • + (lx) k = y n , Publ. Math. Debrecen 91(2017), 369-382. · Zbl 1413.11074 |
| [29] | C. L. Stewart, Primitive divisors of Lucas and Lehmer numbers, in: Transcendence theory: Advances and applications; Proc. Conf. Univ. Cambridge, (A. Baker and D. W. Masser, Eds.), Academic Press, London, 1977, 79-92. · Zbl 0366.12002 |
| [30] | P. M. Voutier, Primitive divisors of Lucas and Lehmer sequences, Math. Comput. 64(1995), 869-888. · Zbl 0832.11009 |
| [31] | G. N. Watson, The problem of the square pyramid, Messenger Math. 48(1919), 1-22. · JFM 46.0213.01 |
| [32] | Z. Zhang, On the Diophantine equation (x − 1) k + x k + (x + 1) k = y n , Publ. Math. Debrecen 85(2014), 93-100. · Zbl 1340.11045 |
| [33] | Z. Zhang, On the Diophantine equation (x − d) 4 + x 4 + (x + d) 4 = y n , Int. J. Numb. Theo. 13(2017), 2229-2243. · Zbl 1392.11018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
