Superelliptic equations arising from sums of consecutive powers
Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^ 3+ x^
3+(x+ 1)^ 3= z^ 2$ in integers $ x $, $ z $. The generalization $(x-1)^ k+ x^ k+ (x+ 1)^ k= z^ n …
3+(x+ 1)^ 3= z^ 2$ in integers $ x $, $ z $. The generalization $(x-1)^ k+ x^ k+ (x+ 1)^ k= z^ n …
[HTML][HTML] On the Diophantine equation (x+ 1) k+(x+ 2) k+...+(2x) k= yn
In this work, we give upper bounds for n on the title equation. Our results depend on
assertions describing the precise exponents of 2 and 3 appearing in the prime factorization …
assertions describing the precise exponents of 2 and 3 appearing in the prime factorization …
Perfect powers that are sums of squares in a three term arithmetic progression
A Koutsianas, V Patel - International Journal of Number Theory, 2018 - World Scientific
Perfect powers that are sums of squares in a three term arithmetic progression Page 1
International Journal of Number Theory Vol. 14, No. 10 (2018) 2729–2735 c World Scientific …
International Journal of Number Theory Vol. 14, No. 10 (2018) 2729–2735 c World Scientific …
[PDF][PDF] Perfect powers that are sums of consecutive squares
V Patel - arXiv preprint arXiv:1707.06678, 2017 - arxiv.org
arXiv:1707.06678v1 [math.NT] 20 Jul 2017 Page 1 arXiv:1707.06678v1 [math.NT] 20 Jul 2017
PERFECT POWERS THAT ARE SUMS OF CONSECUTIVE SQUARES VANDITA PATEL …
PERFECT POWERS THAT ARE SUMS OF CONSECUTIVE SQUARES VANDITA PATEL …
On perfect powers that are sums of cubes of a three term arithmetic progression
A Argáez-García, V Patel - arXiv preprint arXiv:1711.06407, 2017 - arxiv.org
Using only elementary arguments, Cassels and Uchiyama (independently) determined all
squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result …
squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result …
On the Diophantine equation
Z Zhang - International Journal of Number Theory, 2017 - World Scientific
On the Diophantine equation Page 1 International Journal of Number Theory Vol. 13, No. 9 (2017)
2229–2243 c World Scientific Publishing Company DOI: 10.1142/S1793042117501214 On …
2229–2243 c World Scientific Publishing Company DOI: 10.1142/S1793042117501214 On …
Power values of power sums: a survey
Research on power values of power sums has gained much attention of late, partially due to
the explosion of refinements in multiple advanced tools in (computational) number theory in …
the explosion of refinements in multiple advanced tools in (computational) number theory in …
The Diophantine Equation $(x+ 1)^ k+ (x+ 2)^ k+\cdots+ (\ell x)^ k= y^ n $ Revisited
Let $ k,\ell\geq2 $ be fixed integers and $ C $ be an effectively computable constant
depending only on $ k $ and $\ell $. In this paper, we prove that all solutions of the equation …
depending only on $ k $ and $\ell $. In this paper, we prove that all solutions of the equation …
[HTML][HTML] On the equation 1k+ 2k+⋯+ xk= yn for fixed x
We provide all solutions of the title equation in positive integers x, k, y, n with 1≤ x< 25 and
n≥ 3. For these values of the parameters, our result gives an affirmative answer to a related …
n≥ 3. For these values of the parameters, our result gives an affirmative answer to a related …
The equation
MA Bennett, A Koutsianas - arXiv preprint arXiv:2006.10349, 2020 - arxiv.org
In this paper, we solve the equation of the title under the assumption that $\gcd (x, d)= 1$
and $ n\geq 2$. This generalizes earlier work of the first author, Patel and Siksek [BPS16] …
and $ n\geq 2$. This generalizes earlier work of the first author, Patel and Siksek [BPS16] …