Prime Number Brain Dump

20 Sep 2025

I was watching an episode of Futurama titled “The Numberland Gap”. In the episode there was an interaction:

Is it true that every even number is the addition of two primes? … Can you prove it?¹

While Futurama is a meant be numerous cartoon, the writers happen to be very knowledgable. So I thought perhaps this is a true theory that hasn’t been proven. This idea got the squirrel in my head spinning.

I decided it would be interesting to capture what little I know of prime numbers and pull on this idea some on my own. Perhaps afterward I could start to look up some things and see if I gain any insight.

This post will be more or less a brain dump of what I know about primes and pulling on the thread of the assumption that every even number is the addition of two prime numbers.

What Are Prime Numbers

My understanding of prime numbers is that they are whole numbers whose only factors are themselves and one. Where factors can only be whole numbers themselves.²

Let’s look at the first 6 integers:

Number	Factors
1	1
2	1, 2
3	1, 3
4	1, 2, 4
5	1, 5
6	1, 2, 3, 6

Focusing in on 6 we can see:

\[\begin{align} 1 \times 6 &= 6 \\ 2 \times 3 &= 6 \end{align}\]

Finding Primes

I recall in school we had a math exercise where we had 10x10 grid of all the numbers from 1-100. The exercise went something like:

1. Starting at number one.
2. Move to the next number that hasn’t been marked off. This is \(n\).
3. Step by \(n\) and mark off the number you land on.
4. Repeat step \(3\) until the end.
5. Go back to number \(n\).
6. Repeat from step \(2\), until \(n\) is the last unmarked number.

We’ll do an example looking at the first \(20\) numbers. We start with all the numbers unmarked.

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From number \(1\) we move to the next number that hasn’t been marked off, \(2\).

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Now we step every \(2\) and mark off those numbers. One may notice this is all the even numbers.

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We move to the next number that hasn’t been marked off, \(3\).

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Now we step every \(3\) and mark off those numbers. The numbers \(6\), \(12\), and \(18\) were already marked off, but it doesn’t change the process.

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We move to the next number that hasn’t been marked off. This is \(5\). This is the first time we’ve moved past a marked off number.

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Now we step every \(5\) and mark off those numbers. All of these happened to be marked off already.

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We move to the next number that hasn’t been marked off, \(7\).

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Now we step every \(7\) and mark off those numbers. Again all of these happened to be marked off already.

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We move to the next number that hasn’t been marked off, \(11\). If we try to step by \(11\) we would go beyond the end of our max of \(20\).

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In the instructions I wrote:

Step by \(n\) and mark off the number you land on.

This is the same as saying mark off every number that is a multiple of \(n\), that isn’t \(n\). Thinking about it this way one can see that using this method to find all primes in \(1, 2, 3, \ldots, x\), a person only needs to do this up to \(\frac{x}{2}\). That’s why above I stopped at \(11\), because \(\frac{20}{2}=10\)

Assuming Every Equal Number is the Sum of Two Primes

With my little understanding of prime numbers captured I wanted to start thinking about the theory that every even number is the sum of two primes.

An even number is any number which is divisible by \(2\). This means for any prime number, \(p\), we can take \(2p\) and that will form an even number. We can say that \(2p\) is the sum of two primes, because \(2p = p + p\).

With that in hand, my mind immediately went to the idea that the next even number after \(2p\) is \(\left(2p + 2\right)\). By replacing the multiplication with addition and applying the associative property we can say the next even number is \(\left(p + p + 2\right)\).

\[\begin{align} 2p + 2 &= \left(p + p\right) + 2 \\ &= p + \left(p + 2\right) \end{align}\]

We could take the simple assumption that \(\left(p+2\right)\) is prime. Let’s see with the first few primes.

\(p\)	\(\left(p + 2\right)\)
1	3
2	4
3	5
5	7
7	9
11	13
13	15
17	19
19	21

Skipping over \(2\), we don’t get too far until we hit a prime number where \(2\) more isn’t prime. This happens at \(7\) where \(2\) more is \(9\) and \(9\) happens to have the factors \(1, 3, 9\).

We do have some where \(p + 2\) is prime. If we assume that all even numbers are the sum of two primes then we should be able to say that there is at least one prime number, \(q\)³, which is greater than \(p\) and less than \(\left(2p + 2\right)\).

\[\{q : p < q < \left(2p+2\right)\}\]

At this point I realized I was making an assumption, that we knew all the prime numbers up to \(p\). In the diagram above, \(r\) represents one of these previously known prime numbers.

I was curious if we could simply try \(\left(2p + 2\right) = 1 + \left(2p + 1\right)\). \(1\) is prime and \(\left(2p + 1\right)\) is odd.

\(p\)	\(\left(2p + 1\right)\)
1	3
2	5
3	7
5	11
7	15
11	23
13	27
17	35
19	39

This simple approach fizzles out again at \(7\). \(\left(2 \times 7\right) + 1 = 15\) which has factors \(1, 3, 5, 15\).

I was also curious if we could find a pattern in the occurrence of the primes. If we look at how far a prime is from the next one, the delta in the table can we find any obvious patterns?

prime	1	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
delta	n/a	1	1	2	2	4	2	4	2	4	6	2	6	4	2	4	6	6	2	6	4	2	6	4	6	8

It would likely require looking at quite a bit more primes past \(100\) to be able to reach a conclusion about if the delta between primes can provide anything meaningful.

To be honest I’m a bit surprised the primes are that close together up to \(100\). I would have thought they would start to spread out faster and that having primes \(2\) apart wouldn’t happen as often.

Summary

This is about as far as I got with my limited knowledge. I know that large prime numbers are used in things like RSA encryption. I also understand to generate an RSA key pair requires one to search for two large primes and that there are algorithms to search for primes. I don’t know if these algorithms are based on the theory that all even numbers are the sum of two primes. Is that even a theory?

I think I’ll start to do some digging into primes and see what comes out of it. Hopefully not too much is over my head so I can actually learn something.

Footnotes: