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

Argáez-García, Alejandro; Patel, Vandita

doi:10.1016/j.jnt.2020.04.020

Cited by 5 publications

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that since g is contained in A, d w (g) ≤ 0 for all points w S. Therefore, the assumption that d w (g) ≤ 0 implies that g has no poles and thus must be a constant function. This proves part (1).…”

Section: Lemma 23 Let F and G Be A Functions In A And W Be A Point In...supporting

confidence: 63%

See 1 more Smart Citation

Diophantine Equations of the Form Over Function Fields

Ray

2023

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

Let $\ell $ and p be (not necessarily distinct) prime numbers and F be a global function field of characteristic $\ell $ with field of constants $\kappa $ . Assume that there exists a prime $P_\infty $ of F which has degree $1$ and let $\mathcal {O}_F$ be the subring of F consisting of functions with no poles away from $P_\infty $ . Let $f(X)$ be a polynomial in X with coefficients in $\kappa $ . We study solutions to Diophantine equations of the form $Y^{n}=f(X)$ which lie in $\mathcal {O}_F$ and, in particular, show that if m and $f(X)$ satisfy additional conditions, then there are no nonconstant solutions. The results apply to the study of solutions to $Y^{n}=f(X)$ in certain rings of integers in $\mathbb {Z}_{p}$ -extensions of F known as constant $\mathbb {Z}_p$ -extensions. We prove similar results for solutions in the polynomial ring $K[T_1, \ldots , T_r]$ , where K is any field of characteristic $\ell $ , showing that the only solutions must lie in K. We apply our methods to study solutions of Diophantine equations of the form $Y^n=\sum _{i=1}^d (X+ir)^m$ , where $m,n, d\geq 2$ are integers.

show abstract

Section: Lemma 23 Let F and G Be A Functions In A And W Be A Point In...supporting

confidence: 63%

“…We derive conditions for there to be no nonconstant solutions. Given a natural number N, denote by h F [N] the cardinality of the N-torsion in the class group of F. The understanding here is that h F [1] = 1. THEOREM 1.1 (Theorem 2.4).…”

Section: Introductionmentioning

confidence: 99%

Diophantine Equations of the Form Over Function Fields

Ray

2023

Bull. Aust. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…He proved that if k ≥ 1 and n ≥ 2 are fixed, then (1.1) has only finitely many solutions except for the cases (k, n) ∈ {(1, 2), (3,2), (3,4), (5,2)}. In the same paper Schäffer stated the following conjecture on the integral solution of (1.1).…”

Section: Introductionmentioning

confidence: 97%

“…Conjecture 1. [Schäffer, [34]] Let k ≥ 1, n ≥ 2 be integers and (k, n) / ∈ {(1, 2), (3,2), (3,4), (5,2)}. The equation…”

Section: Introductionmentioning

confidence: 99%

“…Koutsianas and Patel [27] completely solved the equation (1.4) for k = 2 and for all values of 1 ≤ d ≤ 5000, using the characterization of primitive divisors in Lehmer sequences by Bilu-Hanrot-Voutier [11]. For 1 ≤ d ≤ 10 6 and k = 3, Argáez-García and Patel [2,3] and Argáez-García [1] studied the equation (1.3) for r = 3, 7 and r = 5, respectively. For k = 4 the equation (1.4) was solved by Zhang [43] for some particular choices of d and Langen [29] under the assumption gcd(x, d) = 1.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation