Prime Matrix

Algebra Level pending

A prime matrix is the numbers listed up to a prime (I just made this up because I am like that...). The general form is the function

$f(M(f(P_n))$

where

$f(M)$

defines the matrix and

$f(P_n)$

is equivalent to the prime-counting function

$\pi(x)$

The matrix is below for $\pi(7)$ :

$\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6\\ 7 \end{bmatrix}$

The question is this:

What's the probability that for any prime matrix, the matrix will always have twin primes?

Example:

For the matrix above, there are $2$ pairs:

$3, 5$ and $5, 7$

Out of $7$ numbers, there are two pairs, therefore the probability is $\frac{2}{7}$ .

\frac{n}{\pi(x)}

\frac{2}{\infty}

\frac{tp_n}{\pi(x)}

\frac{2}{\pi(x)}

1 solution

Yajat Shamji
Aug 31, 2020

Denote the number of pairs of twin primes as $tp_n$

Denote the prime matrix as the prime-counting function (since the prime matrix is, in essence, a visual representation of the prime-counting function).

Therefore the probability is

$\frac{tp_n}{\pi(x)}$

Prime Matrix

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