Aren't there obvious patterns in the primes that no one makes use of and what about this...

Question

Let's take the sequence of naturals at or above two ($2, 3, 4, \dotsc$) and cross out just the primes $2$ and $3$, as well as all their multiples: $$\require{cancel}\cancel{2}, \cancel{3}, \cancel{4}, 5, \cancel{6}, 7, \cancel{8}, \dotsc.$$ If we continue this sequence forever, a pattern emerges, namely $\bullet\bullet\bullet \circ \bullet \circ $ (and repeat).

Such patterns don't exist just for $\{ 2, 3 \}$. Perform this operation for the first $k$ primes for some small $k$ and you'll get a similar pattern (though it will have a very large period). These patterns are shown below are for the $\{ 2,3 \}$ case and then the $\{ 2,3,5 \}$ case respectively. I placed them on a circle to emphasize the fact that they repeat.

Also, some possibly interesting symmetries are immediately apparent. In the first picture, I've marked the modular arithmetic congruence classes. To interpret these pictures, start at node $6x + 2$ on the graph and at $2$ in your enumeration of the naturals. Now traverse the graph in the direction of the arrows as you count up in your naturals. If you're on a black node, then you're a case prime $2,3$ (or $2,3,5$) or a composite. If you land on a white node then you might or not be prime, but on the first cycle it's guaranteed that you are prime. Notice that some twin primes are shown. I don't know if this holds for values of $k > 5$.

I inverted the colors in the second picture to make the symmetry more apparent. But look at the pattern that is present: you get from the first diagram to the second by iterating through the pattern in the first diagram! I refer to the repeating $\bullet\bullet\bullet\circ\bullet\circ$ pattern, except also cross out the added prime $5$ and all its powers. Does this pattern continue, and can it be used to prove the twin prime conjecture?

Answer 1

This is an example of a sieve. It is well known, and I used it over forty years in a program to (iirc) check for primality.

Yes, the pattern continues.

No, it cannot be used to prove the twin prime conjecture. However, in 1919, Viggo Brun used a much more sophisticated sieve to prove that the sum of the reciprocals of the twin primes converges.

Please study the literature before making statements such as "Aren't there obvious patterns in the primes that no one makes use ...". In my opinion, a statement like this makes you sound like a crank.

Answer 2

$2,3,5,7$ case done linearly to fit page: Remember, a $\circ$ is placed over $2,3,5,7$ or any multiple: $$ \begin{matrix} \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \circ & \bullet & \circ & \bullet & \circ & \circ & \circ &\bullet &\circ &\bullet &\circ &\circ & \circ &\bullet &\circ &\circ &\circ &\circ &\circ &\bullet &\circ &\bullet \\ 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 & 31 \\ \end{matrix} $$ now observe the same pattern in the previous case here (except cross out powers of $7$): $$ \begin{matrix} 2 & 3 & 4 & 5 & 6 & 7 & \dots \\ \circ & \circ & \circ & \circ & \circ & \bullet & \circ& \circ & \circ & \bullet & \circ & \bullet & \circ & \circ & \circ &\bullet &\circ &\circ &\circ &\circ & \circ &\bullet &\circ &\circ &\circ &\circ &\circ &\bullet &\circ &\bullet \\ 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 & 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 & 51 & 52 & 53 & 54 & 55 & 56 & 57 & 58 & 59 & 60 & 61 \\ \end{matrix} $$ and the pattern should repeat here, and it does: $$ \begin{matrix} 2 & 3 & 4 & 5 & 6 & 7 & \dots \\ \circ & \circ & \circ & \circ & \circ & \bullet & \circ& \circ & \circ & \bullet & \circ & \bullet & \circ & \circ & \circ &\bullet &\circ &\bullet &\circ &\circ & \circ &\bullet &\circ &\circ &\circ &\circ &\circ &\bullet &\circ &\bullet \\ 62 & 63 & 64 & 65 & 66 & 67 & 68 & 69 & 70 & 71 & 72 & 73 & 74 & 75 & 76 & 77 & 78 & 79 & 80 & 81 & 82 & 83 & 84 & 85 & 86 & 87 & 88 & 89 & 90 & 91 \\ \dots \end{matrix}7 $$

Thanks to achille hui in a comment below, the pattern fails at $77$ (the $79$ was a typo). So the rule is to also cross out all multiples of the newly added prime and not just its powers.