Prime Determinant

May 14th, 2019 by Aziz Lokhandwala

Abstract: This theory is about the determinant which always gives prime number as its value.

2D Determinant

Let “a” and “b” are two consecutive prime numbers. Now, number of prime numbers between “3” and "a” must be always even number of prime numbers. Assume “a” is of digit ‘x’ and ‘y’. Then (x + y) must be always even. $$\begin{vmatrix} 3 & b \\ 2 & a \\ \notag \end{vmatrix}$$

Physical Interpretaion

Determinants are simply an area of the parallelogram which is formed by some vectors with Parallelogram method of vectors. These vectors are more specifically known as EigenVectors. These vectors are associated with the linear system of equations. Each vector is associated with a corresponding value which is known as EigenValue. Now if we correlate Eigen theory with Prime Determinant, we will found that we always get an arrangement of vectors forming some parallelogram in space whose area is always a prime number.

3D Determinant

Gauss's Prime number theorem is the best object to study about prime numbers between any two natural numbers. But this theorem can also be used to have a matrix whose determinant is always a prime number. Let “p” be any prime number, Using Prime number theorem we have, $$f(x) ={ x \over lnx}$$ Now, $$f(p-1) - f(3) = A$$ Here, A = even number Now, determinant is as follows:

$$\begin{vmatrix} 2 & -1 & 3 - p \\ 3 & A & 2\\ p & -3 & p-1\\ \notag \end{vmatrix} \pm 1$$ The value of above Determinant is always a prime number.