Difference between revisions of "Twin Prime Conjecture"
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| − | The ''' | + | The '''Twin Prime Conjecture''' is a [[conjecture]] (i.e., not a [[theorem]]) that states that there are [[infinite]]ly many pairs of [[twin prime]]s, i.e. pairs of primes that differ by <math>2</math>. |
| − | == | + | == Failed Proofs == |
=== Using an infinite series === | === Using an infinite series === | ||
| − | One | + | 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 |
| − | Unfortunately, it has been shown that this sum converges to a constant | + | <center><math>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</math></center> |
| + | of the [[reciprocal]]s of twin primes [[diverge]]s, this would imply that there are infinitely many twin primes. Unfortunately, it has been shown that this sum converges to a constant <math>B</math>, known as [[Brun's constant]]. This could mean either that there are [[finite]]ly many twin prime pairs or that they are spaced "too far apart" for that [[series]] to diverge. | ||
| − | This | + | ===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=== | ||
| + | |||
| + | Some nerd called Zhangaik posted a solution here originally. It's wrong. | ||
| + | |||
| + | ==Alternative statements== | ||
| + | |||
| + | One alternative statement of the Twin Prime Conjecture, is that there exists infinitely many natural numbers not of forms: <cmath>6ab+a+b,6ab+a-b,6ab-a+b,6ab-a-b</cmath> with natural number inputs greater than 0. Because, letting <math>n</math> be of one of these forms one of <math>6n\pm 1</math> 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 <math>12m</math> that have goldbach partitions of distance from <math>m</math> of 1. | ||
{{stub}} | {{stub}} | ||
| − | [[Category: | + | [[Category:Conjectures]] |
[[Category:Number theory]] | [[Category:Number theory]] | ||
Latest revision as of 01:27, 1 May 2024
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
Some nerd called Zhangaik posted a solution here originally. It's wrong.
Alternative statements
One alternative statement of the Twin Prime Conjecture, is that there exists infinitely many natural numbers not of forms: ![\[6ab+a+b,6ab+a-b,6ab-a+b,6ab-a-b\]](http://latex.artofproblemsolving.com/3/f/2/3f26a5abbab6f48b0c4ff609cd63cef48ee2086c.png)
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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