Vandita Patel

activity

20152026

most citedPerfect powers that are sums of consecutive cubes

12 citations · 31 across the 12 of their papers we have counts for

collaborators

16 papers

math.NT2026

On a character-twisted analogue of Schäffer's equation

Kálmán Györy, Vandita Patel, Ákos Pintér +1

Let $f$ be a positive integer, and let $χ$ be a primitive quadratic character of conductor $f$ . Let $k$ be a positive integer, and write $B_{k} (χ, X)$ for the $k$ -th Bernoulli polynom…

math.NT2023

Power values of power sums: a survey

Nirvana Coppola, Mar Curcó-Iranzo, Maleeha Khawaja +2

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…

math.NT2023

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

Nirvana Coppola, Mar Curcó-Iranzo, Maleeha Khawaja +2

We study the equation $(x - 4 r)^{3} + (x - 3 r)^{3} + (x - 2 r)^{3} + (x - r)^{3} + x^{3} + (x + r)^{3} + (x + 2 r)^{3} + (x + 3 r)^{3} + (x + 4 r)^{3} = y^{p}$ , which is a natural continuation of previous works carried out b…

math.NT2021

Odd values of the Ramanujan tau function

Michael Bennett, Adela Gherga, Vandita Patel +1

We prove a number of results regarding odd values of the Ramanujan $τ$ -function. For example, we prove the existence of an effectively computable positive constant $κ$ such that if…

math.NT2019

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

Alejandro Argáez-García, Vandita Patel

We prove that the equation $(x - 3 r)^{3} + (x - 2 r)^{3} + (x - r)^{3} + x^{3} + (x + r)^{3} + (x + 2 r)^{3} + (x + 3 r)^{3} = y^{p}$ only has solutions which satisfy $x y = 0$ for $1 \leq r \leq 1 0^{6}$ and $p \geq 5$ prime…

math.NT2019

A Lucas-Lehmer approach to generalised Lebesgue-Ramanujan-Nagell equations

Vandita Patel

We describe a computationally efficient approach to resolving equations of the form $C_{1} x^{2} + C_{2} = y^{n}$ in coprime integers, for fixed values of $C_{1}$ , $C_{2}$ subject to further co…

math.NT2018

Shifted powers in Lucas-Lehmer sequences

Michael Bennett, Vandita Patel, Samir Siksek

We develop a general framework for finding all perfect powers in sequences derived by shifting non-degenerate quadratic Lucas-Lehmer binary recurrence sequences by a fixed integer.…

math.NT2018

Perfect Powers that are Sums of Squares of an Arithmetic Progression

Debanjana Kundu, Vandita Patel

In this paper, we determine all primitive solutions to the equation $(x + r)^{2} + (x + 2 r)^{2} + \dots + (x + d r)^{2} = y^{n}$ for $2 \leq d \leq 10$ and for $1 \leq r \leq 1 0^{4}$ . We make use of a fa…

math.NT2018

Perfect powers that are sums of squares in a three term arithmetic progression

Angelos Koutsianas, Vandita Patel

We determine primitive solutions to the equation $(x - r)^{2} + x^{2} + (x + r)^{2} = y^{n}$ for $1 \leq r \leq 5, 000$ , making use of a factorization argument and the Primitive Divisors Theorem…

math.NT2017★ 10 cited

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

Alejandro Argáez-García, Vandita Patel

Using only elementary arguments, Cassels and Uchiyama (independently) determined all squares that are sums of three consecutive cubes. Zhongfeng Zhang extended this result and dete…

math.NT2017★ 8 cited

Perfect powers that are sums of consecutive squares

Vandita Patel

We determine all perfect powers that can be written as the sum of at most 10 consecutive squares.

math.NT2017

On perfect powers that are sums of two Fibonacci numbers

Florian Luca, Vandita Patel

We study the equation $F_{n} + F_{m} = y^{p}$ , where $F_{n}$ and $F_{m}$ are respectively the $n$ -th and $m$ -th Fibonacci numbers and $p \geq 2$ . We find all solutions under the assumption $n…

math.AG2017★ 1 cited

On the difference between permutation polynomials over finite fields

Nurdagül Anbar, Almasa Oduzak, Vandita Patel +3

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that if $p > (d^{2} - 3 d + 4)^{2}$ , then there is no complete mapping polynomial $f$ in $\Fp[x]$ of degree $d…

math.NT2016

On powers that are sums of consecutive like powers

Vandita Patel, Samir Siksek

Let $k \geq 2$ be even, and let $r$ be a non-zero integer. We show that for almost all $d \geq 2$ (in the sense of natural density), the equation $$ x^k+(x+r)^k+\cdots+(x+(d-1)r)^k=y…

math.NT2016★ 12 cited

Perfect powers that are sums of consecutive cubes

Michael Bennett, Vandita Patel, Samir Siksek

Euler noted the relation $6^{3} = 3^{3} + 4^{3} + 5^{3}$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gen…

math.NT2015

Superelliptic equations arising from sums of consecutive powers

Michael A. Bennett, Vandita Patel, Samir Siksek

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 $(x - 1)^{k} + x^{k} + (x + 1)^{k} = z^{n}$ (with $x$ , $z$ …