I recently noticed a pattern in twin primes. My questions are: does this pattern continue to hold indefinitely, and how would I prove it? Here's the pattern:
For the $n$th prime, there exists exactly $n-2$ twin prime pairs that can be created as follows:
$p_n$ is the $n$th prime,
$P_p$ is a product of individual primes $p_m$ where $1<m<n$
Here's what I've worked up to:
$n=3$ $(p_n=5)$ has $3-2=1$ twin prime pairs $P_p=3$ gives $(15-4,15-2)=(11,13)$
$n=4$ has 2 $P_p=3$ gives $(17,19)$ $P_p=3*5$ gives $(101,103)$
$n=5$ has 3 (29,31), (227,229), (1151,1153)
$n=6$ has 4 (191,193), (269,271), (2141,2143), (2999,3001)
$n=7$ has 5 (659,661), (2801,2803), (4637,4639), (23201,23203), (255251,255253)
Again, my questions are: does this pattern continue to hold indefinitely, and how would I prove it?
P.S. How about a pic of a brute force Mathematica script up to like 20?