Twin Prime Conjecture

The Twin Prime Conjecture is a conjecture (i.e., not a theorem) that states that there are infinitely many pairs of twin primes, i.e. pairs of primes that differ by $2$ .

Failed Proofs

Using an infinite series

One possible strategy to prove the infinitude of twin primes is an idea adopted from the proof of Dirichlet's Theorem. If one can show that the sum

$B=\frac{1}{3}+\frac{1}{5}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+\frac{1}{17}+\frac{1}{19}+\cdots$

of the reciprocals of twin primes diverges, this would imply that there are infinitely many twin primes. Unfortunately, it has been shown that this sum converges to a constant $B$ , known as Brun's constant. This could mean either that there are finitely many twin prime pairs or that they are spaced "too far apart" for that series to diverge.

Yitang Zhang approach

A weaker version of twin prime conjecture was proved by Yitang Zhang in 2013. This version stated that there are infinitely many pairs of primes that differ by a finite number. The number Yitang chose was 7,000,000. Terence Tao and other people have reduced that boundary to 246 more numbers.

Elementary proof

Proof of the Twin Prime Conjecture

Let $p_n$ be the nth prime number $p_1=2,p_2=3,p_3=5$

Let $P_n$ be the first n prime numbers multiplied together $P_1=p_1,P_2=p_1 \times p_2,P_3=p_1 \times p_2 \times p_3$

Arithmetic Progression

$\{mP_n+a\}$ where $a$ in $A_n$ where $a$ is relatively prime to $P_n$ and less than $P_n$ and $0 \leq m < p_{n+1}$

There always exist numbers $a_1$ and $a_2$ in $A_s$ succh that $a_1+2=a_2$ where $s \geq 3$

Base Case $11,13$ in $A_3$

Induction Case

Let $a_1$ and $a_2$ in $A_n$ such that $a_1+d=a_2$ will propagate at least $p_{n+1}-2$ pairs of numbers which differs by $d$ in $A_{n+1}$

There are a total of $p_{n+1}$ elements generated by arithmetic progression $\{mP_n+a_1\}$ and out of all of the generated elements there is unique element $m_1P_n+a_1$ divisible by $p_{n+1}$

There are a total of $p_{n+1}$ elements generated by arithmetic progression $\{mP_n+a_2\}$ and out of all of the generated elements there is unique element $m_2P_n+a_2$ divisible by $p_{n+1}$

When $m_1 \neq m_2$ there are $p_{n+1}-2$ pairs of numbers $(\{mP_n+a_1\},\{mP_n+a_2\})$ differs by $d, a_1+d=a_2$ in $A_{n+1}$

When $m_1 = m_2$ there are $p_{n+1}-1$ pairs of numbers $(\{mP_n+a_1\},\{mP_n+a_2\})$ differs by $d, a_1+d=a_2$ in $A_{n+1}$

Arithmetic Progression

$\{mP_n+a\}$ where $a$ in $A_n$ where a is relatively prime to $P_n$ and less than $P_n$ and $0 \leq m < P_n$

If there exist an element in $\{mP_n+a_1\}$ divisible by $f$ than in $f$ consecutive elements $x \leq m < x+f$ generated by arithmetic progression $\{mP_n+a_1\}$ there exist unique element $m_1P_n+a_1$ divisble by $f$

Proof of twin prime conjecture by contradiction

For there to not exist two prime numbers which differs by $d, a_1+d=a_2$ There must exist a non-prime number for every value of $m, 0 \leq m <P_n$ in either $\{mP_n+a_1\}$ or $\{mP_n+a_2\}$

All non-prime numbers greater than 1 in $\{mP_n+a\}$ where $a$ in $A_n$ where $a$ in relatively prime to $P_n$ and less than $P_n$ and $0 \leq m < P_n$ must be divisible by an odd number $f$ where $3 \leq f \leq P_n-1$

Removing pairs of numbers from $(\{mP_n+a_1\},\{mP_n+a_2\})$ where either $\{mP_n+a_1\}$ or $\{mP_n+a_2\}$ divisible by $f=P_n-1-2o$ where $3 \leq f \leq P_n-1$

Consider $f=P_n-1$ consective elements $x \leq m < x+f$ generated by arithmetic progression $\{mP_n+a_1\}$ Assume there exist $m_1P_n+a_1$ divisible by $f$ it is unique in these consecutive elements.

Consider $f=P_n-1$ consective elements $x \leq m < x+f$ generated by arithmetic progression $\{mP_n+a_2\}$ Assume there exist $m_2P_n+a_2$ divisible by $f$ it is unique in these consecutive elements.

Assume $m_1 \neq m_2$

Assume the remaining $f-2$ pairs not divisible by $f$ are consective.

Taking the remaining consecutive $f-2$ pairs not divisible by $f$ remove pairs divisible by $f-2$

Consider $f=P_n-1-2$ consective elements $x \leq m < x+f$ generated by arithmetic progression $\{mP_n+a_1\}$ Assume there exist $m_1P_n+a_1$ divisible by $f$ it is unique in these consecutive elements.

Consider $f=P_n-1-2$ consective elements $x \leq m < x+f$ generated by arithmetic progression $\{mP_n+a_2\}$ Assume there exist $m_2P_n+a_2$ divisible by $f$ it is unique in these consecutive elements.

Assume $m_1 \neq m_2$

Assume the remaining $f-2$ pairs not divisible by $f$ are consective.

Continue repeating until with all smaller odd numbers $f=P_n-1-2o$ where $o=0,1,2,3,\ldots$ until $f=3$

There must exist a prime number in $\{m_1P_n+a_1\}$ and $\{m_2P_n+a_2\}$ where $m_1=m_2$ and $0 \leq m < P_n$

Therefore there are infinite number of prime numbers which differ by 2.

Alternative statements

One alternative statement of the Twin Prime Conjecture, is that there exists infinitely many natural numbers not of forms: $6ab+a+b,6ab+a-b,6ab-a+b,6ab-a-b$ with natural number inputs greater than 0. Because, letting $n$ be of one of these forms one of $6n\pm 1$ factors so only if one of variables is 0 will the factorization be trivial (contain only 1 and itself).