On the Diophantine equation (x+ 1) + (x+ 2) + ... + (2x) =y

Bérczes, Attila; Pink, István; Savaş, Gamze; Soydan, Gökhan

doi:10.1016/j.jnt.2017.07.020

Cited by 14 publications

(18 citation statements)

References 17 publications

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“…We shall use the following important results of Schinzel-Tijdeman [23] and Brindza [8] on the superelliptic equation (6) f (x) = y n .…”

Section: Preliminariesmentioning

confidence: 99%

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The Diophantine equation $(x+1)^k+(x+2)^k+\cdots+(\ell x)^k=y^n$ revisited

Bartoli¹,

Soydan²

2020

Publ. Math. Debrecen

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Let k, ℓ ≥ 2 be fixed integers and C be an effectively computable constant depending only on k and ℓ. In this paper, we prove that all solutions of the equation (x + 1) k + (x + 2) k + ... + (ℓx) k = y n in integers x, y, n with x, y ≥ 1, n ≥ 2, k = 3 and ℓ ≡ 1 (mod 2) satisfy max{x, y, n} < C. The case when ℓ is even has already been completed by Soydan (Publ. Math. Debrecen 91 (2017), pp. 369-382).

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“…We shall use the following important results of Schinzel-Tijdeman [23] and Brindza [8] on the superelliptic equation (6) f (x) = y n .…”

Section: Preliminariesmentioning

confidence: 99%

“…He also showed that equation ( 4) has infinitely many solutions with k = 1, 3 and n ≥ 2. In the next year, Bérczes, Pink, Savas and Soydan [6] considered equation ( 4) with ℓ = 2. They proved that it has no solutions if 2 ≤ x ≤ 13, y ≥ 2, and n ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

The Diophantine equation $(x+1)^k+(x+2)^k+\cdots+(\ell x)^k=y^n$ revisited

Bartoli¹,

Soydan²

2020

Publ. Math. Debrecen

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“…We follow Case I in Section 6 of [16] which outlines effective methods to solve equation (6) for fixed values of C 1 , C 2 , p. Theorem 5.1 is crucial to gain good bounds for p. For the convenience of the reader, we shall reproduce the key computational steps used to prove Theorem 1.1 here. For the full exposition and intricate details of computations, see [16].…”

Section: Primitive Prime Divisors Of Lehmer Sequencesmentioning

confidence: 99%

On perfect powers that are sums of cubes of a five term arithmetic progression

Argáez-García

2019

Journal of Number Theory

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We prove that the equation (x − 3r) 3 + (x − 2r) 3 + (x − r) 3 + x 3 + (x + r) 3 + (x + 2r) 3 + (x + 3r) 3 = y p only has solutions which satisfy xy = 0 for 1 ≤ r ≤ 10 6 and p ≥ 5 prime. This article complements the work on the equations (x − r) 3 + x 3 + (x + r) 3 = y p in [2] and (x − 2r) 3 + (x − r) 3 + x 3 + (x + r) 3 + (x + 2r) 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.

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“…x k + (x + r) k + · · · + (x + (d − 1)r) k = y n x, y, d, k, r, n ∈ Z, n ≥ 2. This is still a remarkably active field, with recent results due to [ZB13], [Zha14], [Haj15], [BPS16], [BPS17], [Pat17], [Soy17], [BPSS18], [PS17], [Zha17], [Kou17] and [AGP17]. In this paper, we consider the case d = 3 and k = 2, namely the equation…”

Section: Introductionmentioning

confidence: 99%