New Conjectures on Pythagorean Triples and Biquadratic Equations in the Perfect Cuboid Problem — E8 Intelligence Research

FINDING: The Perfect Cuboid Problem remains unsolved; no perfect cuboid exists with integer edges, face diagonals, and space diagonal, though infinite Euler bricks exist. The arXiv paper (2312.09091v7) proposes new conjectures on Pythagorean triples and biquadratic Diophantine equations related to this. MATH: Perfect cuboid conditions: for edges \(a,b,c\), face diagonals \(dab, dac, dbc\), and space diagonal \(D\): \(a^2+b^2=dab^2\), \(a^2+c^2=dac^2\), \(b^2+c^2=dbc^2\), \(a^2+b^2+c^2=D^2\). Euler brick: first three equations hold (integer solutions exist, e.g., 44, 117, 240). Parametrization of Euler bricks: \(a = 2mn\), \(b = m^2 - n^2\), \(c = 2pq\) with constraints from two Pythagorean triples sharing a leg. The paper conjectures involve biquadratic forms \(x^4 + y^4 = z^2 + w^2\) type constraints, and new families of Pythagorean triples with specific modular restrictions. CONNECTION: The space diagonal condition \(D^2 = a^2+b^2+c^2\) is a 3D Pythago Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com