The Linear Slicing Method for Equal Sums of Like Powers: Modular and Geometric Constraints

TL;DR

This paper investigates the solutions of a specific Diophantine equation under linear constraints, revealing modular obstructions and geometric bounds that limit the existence of solutions for large exponents.

Contribution

It introduces the Modular Divisibility Obstruction (MDO) for shifted slices and combines it with geometric bounds to constrain solutions of the equation.

Findings

The MDO eliminates over 99% of potential solutions for certain exponents.

Unique convexity on the central slice ensures distinct power sums for different pairs.

Solutions are bounded by a logarithmic asymptotic dominance bound related to the exponent.

Abstract

We study the Diophantine equation $a^k + b^k = c^k + d^k$ with integer variables and exponent $k>1$, under the linear constraint $(c+d) - (a+b) = h$. We analyze the geometry and arithmetic of these linear slices. On the central slice $h=0$, we prove strictly convex uniqueness: distinct unordered pairs with the same sum yield distinct power sums. For shifted slices $h\neq 0$, we establish a Modular Divisibility Obstruction (MDO): any solution requires $h$ to be divisible by a specific squarefree modulus $M_k = \prod_{p-1 \mid k-1} p$. This condition creates a strong divisibility filter; for example, if $k=13$, the obstruction eliminates $99.96\%$ of all possible shifts. We combine this arithmetic constraint with a geometric exclusion zone principle and a global overlap bound, showing that the slice size must satisfy $\min\{S, S+h\} \gg |h|$. Finally, we prove an asymptotic dominance bound $k \le \max\{S, S+h\} \log 2$, implying that for any fixed slice, solutions cannot exist for sufficiently large $k$.