# 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 .

## Contents

## 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

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 , 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 be the nth prime number

Let be the first n prime numbers multiplied together

Arithmetic Progression

where in where is relatively prime to and less than and

There always exist numbers and in succh that where

Base Case in

Induction Case

Let and in such that will propagate at least pairs of numbers which differs by in

There are a total of elements generated by arithmetic progression and out of all of the generated elements there is unique element divisible by

There are a total of elements generated by arithmetic progression and out of all of the generated elements there is unique element divisible by

When there are pairs of numbers differs by in

When there are pairs of numbers differs by in

Arithmetic Progression

where in where a is relatively prime to and less than and

If there exist an element in divisible by than in consecutive elements generated by arithmetic progression there exist unique element divisble by

Proof of twin prime conjecture by contradiction

For there to not exist two prime numbers which differs by There must exist a non-prime number for every value of in either or

All non-prime numbers greater than 1 in where in where in relatively prime to and less than and must be divisible by an odd number where

Removing pairs of numbers from where either or divisible by where

Consider consective elements generated by arithmetic progression Assume there exist divisible by it is unique in these consecutive elements.

Consider consective elements generated by arithmetic progression Assume there exist divisible by it is unique in these consecutive elements.

Assume

Assume the remaining pairs not divisible by are consective.

Taking the remaining consecutive pairs not divisible by remove pairs divisible by

Consider consective elements generated by arithmetic progression Assume there exist divisible by it is unique in these consecutive elements.

Assume

Assume the remaining pairs not divisible by are consective.

Continue repeating until with all smaller odd numbers where until

There must exist a prime number in and where and

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: with natural number inputs greater than 0. Because, letting be of one of these forms one of factors so only if one of variables is 0 will the factorization be trivial (contain only 1 and itself).

Another is that there are infinitely many values that have goldbach partitions of distance from of 1.

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