Let A be a hyperdimensional matrix of order n, where n is a positive integer greater than 1. A hyperdimensional matrix is defined as an n-dimensional array of complex numbers, represented as A(i_1, i_2, ..., i_n), where i_1, i_2, ..., i_n are integers ranging from 1 to m for each dimension, and m is a positive integer greater than or equal to 2. Each entry of the matrix A is denoted as A(i_1, i_2, ..., i_n) = x + yi, where x and y are real numbers, and i is the imaginary unit.
The Hyperdimensional Matrix Conjecture states:
For any hyperdimensional matrix A of order n and dimension m, there exists a sequence of n distinct prime numbers, p_1, p_2, ..., p_n, such that the determinant of A can be expressed as:
det(A) = (p_1^(c_1)) * (p_2^(c_2)) * ... * (p_n^(c_n))
Where c_1, c_2, ..., c_n are positive integers determined by the properties of the matrix A and the chosen prime numbers p_1, p_2, ..., p_n.
Prove or disprove the Hyperdimensional Matrix Conjecture, providing a counterexample or a rigorous mathematical proof.
Note: This problem involves advanced concepts from linear algebra, number theory, and abstract algebra. Good luck!