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Section 10.5 The Twin Prime Conjecture

It was relatively easy to prove that there are infinitely many primes (Theorem 10.4.1). In order to come up with a new mathematical result, a great deal of study, investigation, and insight is often required. Ideas arise, steps toward a proof are taken, and sometimes those ideas have to be tweaked. In this process, it is possible to develop a statement that is believed to be true but has not been formally proven. Such a statement is called a conjecture and is often known in mathematics as an open problem. We conclude this section by presenting an important conjecture involving primes. While the statement of the conjecture is easy to understand and computer experiments have not come up with a counterexample, we do not know whether it is true.

We give an overview over the twin prime conjecture in the video in Figure 10.5.1 Further details are given below.

Figure 10.5.1. Twin Prime Conjecture by Matt Farmer and Stephen Steward.

We start with the definition of twin primes.

Definition 10.5.2.

Let \(p\) be an integer. If \(p\) and \(p+2\) are both prime, then \((p,p+2)\) is called a twin prime pair.

The first four twin prime pairs are

\begin{equation*} (3,5), (5,7), (11, 13), (17,19). \end{equation*}

Determine whether or not \(89\) is a part of a twin prime pair.

Solution.

First we check whether \(89\) is a prime number. Checking the list of primes in Table 10.2.4 we find that \(89\) is a prime number. To determine whether \(89\) is part of a twin prime pair we need to check whether one of \(89-2=87\) or \(89+2=91\) is a prime number. Checking Table 10.2.4 we find that neither \(87\) nor \(91\) are prime. Thus \(89\) is not a part of a twin prime pair.

Determine whether or not \(137\) is a part of a twin prime pair.

Solution.

First we check whether \(137\) is a prime number. Checking the list of primes in Table 10.2.4 we find that \(137\) is a prime number. To determine whether \(137\) is part of a twin prime pair we need to check whether one of \(137-2=135\) or \(137+2=139\) is a prime number. Checking Table 10.2.4 we find that neither \(135\) is not a prime number. But we find that \(139\) is prime. Thus \(137\) and \(139\) are twin prime. So \(137\) is part of a twin prime pair.

The Twin Prime Conjecture is the claim that there are infinitely many twin prime pairs.

This is the first (and only) conjecture that you will encounter in this course. It is important to distinguish conjectures and theorems. Both conjectures and theorems are statements. While theorems are true statements, for a conjecture nobody has determined yet whether it is true or false. As soon as it is determined by a proof that a conjecture is true, it becomes a theorem. Also see the treatment of this topic in the preface in Section 0.2.

Decide if each of the following statements is a definition, a theorem or a conjecture.

  1. Let \(n\in\mathbb{N}\) then \(\gcd(n,n+1)=1\text{.}\)

  2. There are infinitelty many twin primes.

  3. For \(n\in\mathbb{N}\) we set \(\mathbb{Z}_n^\otimes:=\lbrace 1,2,3,\dots,n-1\rbrace\text{.}\)

  4. \(p\in\mathbb{N}\) is a prime number means that \(p > 1\) and that the only divisors of \(p\) are \(1\) and \(p\text{.}\)

It is outside the scope of this course to try to prove the twin prime conjecture. Nevertheless it is interesting to see whether twin primes exist (if not the conjecture would be false and not of much interest).

How many twin prime pairs are there up to \(100\) ?

Solution.

With Table 10.2.4 we get that the twin prime pairs up to \(100\) are:

\((3,5)\text{,}\) \((5,7)\text{,}\) \((11, 13)\text{,}\) \((17,19)\text{,}\) \((29,31)\text{,}\) \((41,43)\text{,}\) \((59,61)\text{,}\) and \((71,73)\)

Thus there are eight twin prime pairs up to \(100\text{.}\)

It appears that there are fewer twin primes than there are primes. In Checkpoint 10.5.9 count the number of primes and twin primes up to a given natural number.

The number of primes up to 30 is

The number of twin prime pairs up to 30 is

The number of primes up to 60 is

The number of twin prime pairs up to 60 is

The number of primes up to 90 is

The number of twin prime pairs up to 90 is

The number of primes up to 120 is

The number of twin prime pairs up to 120 is

Answer 1.

\(10\)

Answer 2.

\(4\)

Answer 3.

\(17\)

Answer 4.

\(6\)

Answer 5.

\(24\)

Answer 6.

\(8\)

Answer 7.

\(30\)

Answer 8.

\(10\)

Progress towards proving the twin prime conjecture (Conjecture 10.5.6) has been made recently. In 2013, Yitang Zhang [10] made a major breakthrough by proving that there are infinitely many primes \(p\) and \(q\) such that \(p-q\le 70,000,000\text{.}\)

Soon after this was improved considerably, such that now it is known that there are infinitely many primes \(p\) and \(q\) such that \(p-q\le 246\text{.}\) When it is proven that there are infinitely many primes \(p\) and \(q\) such that \(p-q\le 2\text{,}\) the twin prime conjecture is proven.

We end this section with a song about the twin prime conjecture in Figure 10.5.10.

Figure 10.5.10. The Twin Prime Conjecture by Rob Morsberger (from PBS Nova)