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An Elementary Approach For Sums Of Consecutive Cubes Being Prime Power Squares

Atilla Akkuş

Abstract

This paper proposes an elementary solution to a special case of finding all perfect squares that can be written as sum of consecutive integer cubes. It is shown that there are no non-trivial solutions if the perfect square is a prime power, i.e., it is divisible by two different primes if a non-trivial one exists. Solution mostly depends on $v_{p}(x)$ and general forms of Pythagorean triples.

An Elementary Approach For Sums Of Consecutive Cubes Being Prime Power Squares

Abstract

This paper proposes an elementary solution to a special case of finding all perfect squares that can be written as sum of consecutive integer cubes. It is shown that there are no non-trivial solutions if the perfect square is a prime power, i.e., it is divisible by two different primes if a non-trivial one exists. Solution mostly depends on and general forms of Pythagorean triples.
Paper Structure (11 sections, 1 theorem, 27 equations)

This paper contains 11 sections, 1 theorem, 27 equations.

Table of Contents

  1. Introduction
  2. For $x$ equals one
  3. For $x$ bigger than one
  4. For $k$ even
  5. First Case
  6. Second Case
  7. For $k$ odd
  8. For $q$ less than $t$
  9. First Case
  10. Second Case
  11. For $q$ equals $t$

Key Result

Theorem 1

For $x,k,r \in\mathbb{Z^+}$ and $p$ prime, all solutions to the equation are $(x,k,p,r) = (p^{2c},1,p,3c), (1,2,3,1)$ where $c$ is an arbitrary positive integer.

Theorems & Definitions (2)

  • Theorem 1
  • Remark 1