Proof of the twin prime conjecture, and why the twin primes exist This proof is all about structuring the natural numbers and the rules that logically apply within this structure. First we find the reason why the twin primes exist, and from that we then go on to prove the twin prime conjecture. A nice new trick for the mathematical toolbox is included. Lets start by generating all the natural numbers starting from 2. We begin with 2 numbers, the numbers 2 and 3 in the first column as shown in the figure. Within this first column we denote the prime number 2 with a circle with a "P" inside it and the number 3 with a cross. We then copy this first column 3 times in order to form the second column. We leave the circular space blank in all 3 copies of the first column and only show the now 3 crosses. We then take the first cross and replace it with a circle with a "P" inside it in order to denote the prime number 3. Next we copy the second column 5 times to create the third column and we replace the first cross with a circle with a "P" inside it. We now also replace the cross with the square of 5 (= 25 = a multiple of the prime number 5) with a circle with an "M" in it. I am sure that you get the pattern by now, but just to recap. All the columns are n copies of the previous one, whereby n is the prime number of the column and the columns are in order of the prime numbers. Within this each column we denote the prime number itself with a circle with a "P" inside it and all the multiples of the prime number with a circle with an "M" inside it. The positions with a circle are left blank in all the subsequent columns, while the others are denoted with a cross. Now some simple rules about the numbers that we have just generated are apparent. (1) This system generates al the numbers from 2 to infinity. (2) Each number is generated exactly once. (3) The crosses in each column of a prime number can only be filled by higher prime numbers or their multiples. (4) None of the crosses in a column are divisible by a prime number that is lower or equal to the prime number of the column. Now you will have noticed that the cross of the number 3 in the first column resulted in two crosses in the second column (the numbers 5 and 7). This pair of crosses then gets infinitely copied into all the subsequent columns. In some cases, like the 25 for example, one or two of the crosses gets lost, but many pairs of crosses remain. This copying of the first set of crosses is the reason why the twin primes keep popping up into infinity and thus why the twin primes exist. So the creation of twin primes will go on into infinity, and any formula describing them needs to be a constant plus something divisible by both 2 and 3 as they were formed due to a multiplication with these numbers. Note that each next prime number and its multiples take away crosses in a way that is out of phase and out of sync with the structure created by the previous primes, effectively resulting in a pseudo random spread. The proof of the twin prime conjecture must therefore be equal to proving that there is no mechanism that could totally take out all of the twin primes into infinity. For such a mechanism to work it would have to be both in phase and in sync with the twin primes, which requires that it would have to be divisible by both 2 and 3. And yet such a mechanism would also be required to be comprised solely of prime numbers larger than 3, and by definition none of these will be divisible by either 2 or 3 (let alone both). So the mechanism required simply cannot exist and this therefore proves the twin prime conjecture to be correct (plus al the other conjectures based on repeating patterns within the prime number set that originate from the structure above described). Or to put it in another way, the pseudo random spread we mentioned before is not truly random and one of its features is that it can never be in phase and in sync with the previous prime numbers, so patterns created by the previous prime numbers persist. Finally, it might conceptually be easier to call the positions of the crosses "holes" as these are the holes where future primes and their multiples will fall into place. https://www.academia.edu/60193741