Local Obstructions and Baker-Type Gap Bounds for the Beal Equation

This work presents a reproducible framework for studying primitive solutions to Beal’s equation Ax+By=CzA^x+B^y=C^zAx+By=Cz with gcd⁡(A,B,C)=1\gcd(A,B,C)=1gcd(A,B,C)=1 and x,y,z>2x,y,z>2x,y,z>2. Locally, the method organizes standard congruence information—2-adic valuations, odd-prime lifting-the-exponent (LTE) facts, and small power-residue images—into branch data. Via the Chinese Remainder Theorem, each branch yields a finite modulus ttt and residue set RRR such that any putative solution must satisfy Cz mod t∈RC^z \bmod t \in RCzmodt∈R. A worked example illustrates how common parity/divisibility patterns lead to t=24t=24t=24 with R={0}R=\{0\}R={0}.

Globally, the paper writes Cz=L+UC^z=L+UCz=L+U with L=max⁡{Ax,By}L=\max\{A^x,B^y\}L=max{Ax,By} and applies explicit two-logarithm lower bounds (Baker/Matveev-type) to obtain a universal lower bound on the relative gap U/LU/LU/L in terms of log⁡H\log HlogH (height H=max⁡{∣A∣,∣B∣,∣C∣}H=\max\{|A|,|B|,|C|\}H=max{∣A∣,∣B∣,∣C∣}) and the largest exponent M=max⁡{x,y,z}M=\max\{x,y,z\}M=max{x,y,z}. A key implication is conditional: if an exponential upper bound on the gap is established (e.g., U/L≤e−cMU/L\le e^{-cM}U/L≤e−cM), then one obtains a polylogarithmic cap M≪(log⁡H)4M \ll (\log H)^4M≪(logH)4. The note does not resolve Beal’s conjecture; rather, it packages the local sieve and global gap calculus into a clear, testable toolkit with verification scripts/tables and fixed random seeds.