I thought of this when I thought of the problem, I go into more detail in the link I provide. But first you can prove that a number is the second of a pair of twin primes if and only if it divides no prime up to the square root of the number with a remainder of 2 or 0. Then I use this estimate for the number of pairs of twin primes: up to N=(p+1)^2-1 for a prime p. N*(1/2)*(3/5)*(5/7)*…*(p-2)/p, this follows from the statistical notion that of all the numbers up to N, (3/5) for example will not divide 5 by 0 or 2. so (p-2)/p is the fraction of numbers up to N that does not divide p with a remainder of 0 or 2 and these are multiplied for every prime up to the square root of N to give an estimate of the number of pairs of twin primes less than N. Then I show that this estimate tends towards infinity with increasing N, thus I conclude that there are an infinite number of twin primes. Maybe someone could explain why this doesn’t work?
http://benpaulthurstonblog.blogspot.com/2013/11/even-simpler-idea-for-proof-of-twin.html

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