The Twin Prime Generator

A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43).

(source)

So I realized something interesting the other day, given twin primes \(p, p+2\) and \(q\) the number occurring in between:

Note that \( q \) is always divisible by \( 6 \)! As you can imagine, Dear Reader, I wondered if this held true generally. (And is it turns out, yes. It does!)

Let’s convince ourselves of this.

First, let’s remember that we only care about twin primes > 4. Then, as we showed in a previous post:

Given any three consecutive numbers, exactly one number will be divisible by 3.

^ Just to rehash the above super quick, suppose we have some consecutive number sequence

\( p, q, r, s \)

such that

\( q = p + 1 \), \( r = p + 2 \) and \( s = p + 3 \)

Let’s let \( p \) be divisble by \( 3 \), or in other words \( p = 3n \) where \( n \) is some natural number.

Now, we can represent \( p, q, r, s \) as \( 3n, 3n + 1, 3n + 2, 3n + 3 \).

Let’s look at this set in groups of \( 3 \), eg:

• \( \bold{3n, 3n+1, 3n+2}, 3n + 3\) | here, \( 3n \) is clearly divisible by \( 3 \)
• \( 3n, \bold{3n+1, 3n+2, 3n + 3}\) | here, \( 3n + 3 \) is clearly divisible by \( 3 \)

(And of course, if \( 3n \) is in the middle of the set, so \( 3n -1, 3n, 3n + 1 \), we observe the same result).

All in all, let’s move forward having convinced ourselves that given any set of three consecutive numbers one number will be a multiple of three.

Now,

Given a set of twin primes \( p, p+2 \) (such that \( p \) > 4), we know that there exists a non prime \( p + 1 \) in between and that by definition \( p \) and \( p + 2 \) cannot be divisible by \( 3 \) (because they are prime). Therefore, \( p + 1 \) must be divisible by 3 (as per our conclusion above)

Also:

Given a set of twin primes \( p, p+2 \) (such that \( p \) > 4), we know that both \(p\) and \(p+2\) must be odd. Therefore, \(p+1\) must be even.

For this reason, we can conclude that:

• \(p+1\) is even (it is divisible by 2)
• \(p+1\) is divisible by 3

And so, \(p+1\) is also be divisible by \(6\). For this reason:

Any number \(q\) occurring between twin primes \(p\) and \(p+2\) must be a multiple of \(6\).

Ta-da 🎉

And finally, if we were to continue our table above (but truncated, because lazy)

We arrive at a number sequence:

$$ p(x) = 1,2,3,5,7,10,12,17,18,23,25,30,32… $$

that we can think of as a twin prime generator.

Basically, provided we have some \(x\) such that we can evaluate \(p(x)\), we are guaranteed to find \(x\)th twin prime pair at \((p(x)-1, p(x)+1) \).

I’m not yet sure how \(p(x)\) can be computed though (for now at least - though it is likely that no such computation exists). Will report back any breakthroughs if/as they occur 🙏

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