Beal Conjecture Proof & Beautiful Mathematics
A. A. Frempong · viXra · 2021
The symmetric structure of the Beal conjecture proof, in this paper, exemplifies the beauty in mathematics. The author applies basic mathematical principles to surely, instructionally, and beautifully, prove the original Beal conjecture that if A^x + B^y = C^z, where A, B, C, x, y, z are positive integers and x, y, z > 2, then A, B and C have a common prime factor. One will let r, s, and t be prime factors of A, B and C, respectively, such that A = Dr, B = Es, C = Ft, where D, E, and F are positive integers. Then, the equation A^x + B^y = C^z becomes D^xr^x + E^ys^y = F^zt^z. The proof would be complete after showing that the equalities, r^x = t^x, s^y = t^y and r = s = t, are true. The proof of the above equalities would involve showing that the ratios, (r^x)/(t^x) = 1 and (s^y)/(t^y) =1, which would imply that r = s = t. The main principle for obtaining relationships between the prime factors on the left side of the equation and the prime factor on the right side of the equation is that the power of each prime factor on the left side of the equation equals the same power of the prime factor on the right side of the equation. High school students can learn and prove this conjecture for a bonus question on a final class exam.