Geometry of terms in Beal's Conjecture

- Ramanraj K

31^st May, 2020

Beal's Conjecture states that if A^x + B^y = C^z, where A, B, C, x, y and z are positive integers and x, y and z are all greater than 2, then A, B and C must have a common prime factor. The geometry of the terms are drawn here using x3d.

The equation in the Conjecture is derived directly from the identity A^x + (A-1)A^x = A^(x+1) where A=2, and in all other cases where A>2, may be derived directly or by repeated substitution. It may also be noted that A, B and C must have a common prime factor or a set of common prime factors, and the greatest prime factor must be common. Further, at least one of the terms A, B and C would have either 2 to 3 as a prime factor. The common prime factors are divided by their cubes to set the scale of the subcube, and the remainder is used to render the geometry of the terms. Pythagoras, Fermat, Beal, and Galois are all on the same page here. Scroll down to view the visualization.

3³ + 6³ = 3⁵ = 27 + 216 = 243

Common Prime factors of 27, 216, 243: 3 3 3 = 3³

Greatest Common Prime factor of 27, 216, 243: 3

TERM #1: 3³ = 27
SCALE 1:3 Length: 1 Width: 1 Height: 1

TERM #2: 6³ = 216
SCALE 1:3 Length: 2 Width: 2 Height: 2

TERM #3: 3⁵ = 243
SCALE 1:3 Length: 3 Width: 1 Height: 3

Note: If the length, width or height is too large, it may not be rendered correctly.