Proving theorems and special cases (Part 7): The twin prime conjecture

In a recent class with my future secondary math teachers, we had a fascinating discussion concerning how a teacher should respond to the following question from a student:

Is it ever possible to prove a statement or theorem by proving a special case of the statement or theorem?

Usually, the answer is no. In this series of posts, we’ve already seen that a conjecture could be true for the first 40 cases or even the first 10^{316} cases yet ultimately prove false for all cases.

For the next few posts, I thought I’d share a few of the most famous unsolved problems in mathematics… and just how much computational work has been done to check for a counterexample.

2. The twin prime conjecture (see here and here for more information) asserts that there are infinitely many primes that have a difference of 2. For example:

3 and 5 are twin primes;

5 and 7 are twin primes;

11 and 13 are twin primes;

17 and 19 are twin primes;

29 and 31 are twin primes; etc.

While most mathematicians believe the twin prime conjecture is correct, an explicit proof has not been found. The largest known twin primes are

3,756,801,695,685 \times 2^{666,669} \pm 1,

numbers which have 200,700 decimal digits. Also, there are 808,675,888,577,436 twin prime pairs less than 10^{18}.

Most mathematicians also believe that there are infinitely many cousin primes, which differ by 4:

3 and 7 are cousin primes;

7 and 11 are cousin primes;

13 and 17 are cousin primes;

19 and 23 are cousin primes;

37 and 41 are cousin primes; etc.

Most mathematicians also believe that there are infinitely many sexy primes (no, I did not make that name up), which differ by 6:

5 and 11 are sexy primes;

7 and 13 are sexy primes;

11 and 17 are sexy primes;

13 and 19 are sexy primes;

17 and 23 are sexy primes; etc.

(Parenthetically, a “sexy” prime is probably the most unfortunate name in mathematics ever since Paul Dirac divided a bracket into a “bra” and a “ket,” thereby forever linking women’s underwear to quantum mechanics.)

While it is unknown if there are infinitely many twin primes, it was recently shown — in 2013 — that, for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N. In 2014, this upper bound was reduced to 246. Furthermore, if a certain other conjecture is true, the bound has been reduced to 6. In other words, there are infinitely many twin primes or cousin primes or sexy primes… but, at this moment, we don’t know which one (or ones) is infinite.

In November 2014, Dr. Terence Tao of the UCLA Department of Mathematics was interviewed on the Colbert Report to discuss the twin prime conjecture… and he does a good job explaining to Stephen Colbert how we can know one of the three categories is infinite without knowing which category it is.

From the Colbert Report:






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