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

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 $k$.