It is well-known that using the sieve of Eratosthenes, we can generate the sequence of primes $2,3,5,7,11,13,17,19,\ldots\), which is known to be infinite. It is a longstanding problem to prove if there are (or there are not) infinitely many twin primes $$(3,5),~(5,7),~(11,13),~(17,19),~(29,31),\ldots$$. There exists an efficient method[^6913] to sieve all twin primes of the form $$(6k-1,6k+1)$$, $$k=1,2,\ldots$$. Note that these are all twin primes except the twin primes $$(3,5)$$. This is because even almost all primes (and not only twin primes) can be written in this form1. ### A Sieve for Twin Primes and the Twin Prime Sequence 1. Start with the sequence of positive natural numbers $$1,2,3,4,\ldots$$. 2. Use the known infinite sequence of primes $$p > 3$$, i.e. the primes $$5, 7, 11, 13,17, 19, 23, 29,\ldots$$ in the following way: # 1. Set the number $$f_p$$ to the value of the floor function $$f_p:=\lfloor (p+1)/6\rfloor$$. # 1. Sieve from the above sequence of all positive natural numbers the numbers $$pk-f_p$$ and $$pk+f_p$$ for $$k=1,2,3,\ldots$$. # 1. Repeat this procedure for all primes $$p > 3$$. 1. The remaining sequence starts with $$k=1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33,\ldots$$. 2. The twin primes can be restored from this remaining sequence as follows: # 1. Build pairs of numbers $$(6k-1,6k+1)$$ for $$k=1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33,\ldots$$. # 1. These pairs are all twin primes, except $$(3,5)$$, i.e. the twin primes $$(5,7),~(11,13),~(17,19),~(29,31),~(41,43),\ldots$$. The following figure visualizes this sieve method: With this respect, the sequence $$k=1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33,\ldots$$ can be called the twin prime sequence. The following lemma formalizes this result. # Lemma: Sieve for Twin Primes Let $n\ge 1$be an integer. The integer$a_n:=36n^2-1$is the product of two twin primes$p=6n-1$and$q=6n+1$if and only if$n\not\equiv\pm f_s\mod s$for all primes$s$with$5\le s\le\sqrt q,$where, using the floor function, the residue classes$f_s\$ are defined by $$f_s:=\left\lfloor\frac {s+1}6\right\rfloor.$$

 Table of Contents Proofs: 1 Thank you to the contributors under CC BY-SA 4.0! Github: References Bibliography Piotrowski, Andreas: "Anmerkungen zur Verteilung der Primzahlzwillinge", Master’s thesis, Frankfurt am Main, 1999 Footnotes This is because for all remaining natural numbers $$n$$, there is a $$k\ge 1$$ such that $$n=6k-2$$, or $$n=6k+2$$, or $$n=6k-3$$, or $$n=6k+3$$, and all these numbers are divisible by $$2$$ and $$3$$. Since they are composite (i.e. not prime), all the remaining primes (and twin primes) must be of the form $$n=6k-1$$ or $$n=6k+1$$. ↩ 
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