The Diophantine equation (x+1)k + (x+2)k + ... plus (lx)k = yn revisted

被引：5

作者：

Bartoli, Daniele ^{[1
]}

Soydan, Gokhan ^{[2
]}

机构：

[1] Univ Perugia, Dept Math & Informat, Perugia, Italy

[2] Bursa Uludag Univ, Dept Math, TR-16059 Bursa, Turkey

来源：

PUBLICATIONES MATHEMATICAE DEBRECEN | 2020年 / 96卷 / 1-2期

关键词：

Bernoulli polynomials; high degree equations; PERFECT POWERS; SUMS;

D O I：

10.5486/PMD.2020.8604

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

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 not equal 3 and l 1 (mod 2) satisfy max{x, y, n} < C. The case when is even has already been completed by the second author (see [24]).

引用

页码：111 / 120

页数：10

共 14 条

[1] On the Diophantine equation (x+1)k (x+2)k + . . . plus (lx)k = yn
Soydan, Gokhan
PUBLICATIONES MATHEMATICAE-DEBRECEN, 2017, 91 (3-4): : 369 - 382
[2] On the Diophantine equation (x+1)k + (x+2)k + ... + (2x)k = yn
Berczes, Attila
Pink, Istvan
Savas, Gamze
Soydan, Gokhan
JOURNAL OF NUMBER THEORY, 2018, 183 : 326 - 351
[3] On the Diophantine equation 1k+2k +...+ xk = yn
Bennett, MA
Gyory, K
Pintér, A
COMPOSITIO MATHEMATICA, 2004, 140 (06) : 1417 - 1431
[4] On the equation 1k+2k + ... + xk=yn
Györy, K
Pintér, A
PUBLICATIONES MATHEMATICAE-DEBRECEN, 2003, 62 (3-4): : 403 - 414
[5] On the number of solutions of the equation 1k+2k+•••+(x-1)k=yz
Brindza, B
Pintér, A
PUBLICATIONES MATHEMATICAE-DEBRECEN, 2000, 56 (3-4): : 271 - 277
[6] On some generalizations of the diophantine equation s(1k+2k + ... + xk) + r = dyn
Rakaczki, Csaba
ACTA ARITHMETICA, 2012, 151 (02) : 201 - 216
[7] On a conjecture of Schaffer concerning the equation 1k + ... + xk = yn
Hajdu, L.
JOURNAL OF NUMBER THEORY, 2015, 155 : 129 - 138
[8] On the diophantine equation x2+(p1Z1 ••• psZs)2=2yn
Pink, I
PUBLICATIONES MATHEMATICAE-DEBRECEN, 2004, 65 (1-2): : 205 - 213
[9] On the solutions of the Diophantine equation Fn ± a(10m-1)/9 k!
Adedji, Kouessi Norbert
Luca, Florian
Togbe, Alain
JOURNAL OF NUMBER THEORY, 2022, 240 : 593 - 610
[10] A Probabilistic Look At 1k+2k + MIDLINE HORIZONTAL ELLIPSIS plus nk
Farhadian, Reza
RESONANCE-JOURNAL OF SCIENCE EDUCATION, 2023, 28 (03): : 479 - 482

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