The first lemma will be: "The second of a pair of twin primes does not have a remainder of 2 when divided by any prime up to its square root, and a number is the second of a pair of twin primes if it doesn't have a remainder of 2 or 0 when divided by any prime up to it's square root."
Proof: Suppose P did divide a prime p less than it's square root with a remainder of 2. Then the number 2 less than P would divide it exactly. Thus the number 2 less than P is not prime, which contradicts the fact that P was the second of a pair of twin primes. Also if P does not divide any number less than it's square root with a remainder of 0 then it is prime, and if it also does not divide any of those with a remainder of 2 then the number 2 less than P does not divide them either, which means it is also prime. Hence, P is the second of a pair of twin primes.
Theorem: Let N = (p+1)^2-1, a way to roughly count the number of primes less than N is:
Proof: Roughly 1/2 of all the numbers up to N won't divide 2, 2/3 won't divide 3, and so on up to p-1. Whether a number divides p or divides another prime q happen independently of one another so the percentages are multiplied. (p+1)^2 is also the first number where factors of p+1 need to be considered as any number that divides p+1 evenly before that also divides a smaller prime.
Theorem 2: Let N = (p+1)^2-1, a way to count one of each of every pair of twin primes less than N is roughly:
N*(1/2)*(1/3)*(3/5)...*(p-1)/p: The same as the formula from theorem 1 but the numerators of all but the 1/2 term are decreased by 1.
Proof: Because of the lemma, we know that numbers that have a non-zero remainder when divided by every p up to the square root, and also not 2 when divided by each p are one of a pair of twin primes. For every p but 2 this is one less possible remainder so every term other than 1/2's numerator is reduced by 1. For example there are 3 possibilities for the remainder when divided by 5, 1,3, and 4. So 3/5 appears in the fraction. This formula thus roughly counts the number of twin primes less than N.
Theorem 3: There are an infnite number of twin primes estimated by the formula in Theorem 2:
Proof: Suppose the number of twin primes up to N used theorem 2's formula and counted P pairs of twin primes by:
for now we can write that as:
We can show that if we replace p with the next bigger prime q, the number of twin primes estimated always increases because the ratio of estimated primes is:
This ratio is always greater than 1 as p gets larger because the ratio of q^2-1 to p^2-1 is larger than 1 by more than the ratio of (q-2) to q is smaller than 1. This means there must be a positive number of estimated pairs of twin primes between p^2-1 and q^2-1 And since there are an infinite number of primes to choose p and q from there must be an infinite number of pairs of twin primes...