KILLING PRIMES

A special delivery from THE SET THEORY GUIDE FOR ARTISTS

                        Genre of math: Horror/Comedy/Action/Adventure/Romance

INTRODUCTION TO M

Let us go into an unusual, yet familiar, world of arithmetic, a world built from prime blocks. First, we need to find a portal into this world. Oh wait, I see one. Do you see the mind of that mathematician sitting in the lobby?

That’s a portal. Want to go? Let’s goooooooooo!

In M, instead of numbers like we have, there are structures made of prime blocks. We will start building very soon. How can you make 4 with the blocks above (not using the 1-block)? How about 5? While you are thinking of building the next numbers, let’s also think about prime numbers. What makes primes so special and intriguing? Prime numbers have two major features: they are creative and they are unique. Primes are unique because a prime number p > 1 has only two positive integer factors, namely 1 and p. Primes are also very creative. The fundamental theorem of arithmetic says that every positive integer can be written as a product of primes. Thus, the creativity of the primes gives us of all the rest of the natural numbers. In M, we see what happens when we pause or mute the creative ability of a prime number.

In M, we have the absence of blocks, the nothing block, which we interpret as 0. Let us say “our world” or V when we are referring to the way we usually see numbers and perform arithmetic. So, the nothing block in M is interpreted as 0 in our world. Then, in M, we are given ☐, which we would interpret as 1 in V. Just like in our world, we will not think of ☐ as a prime block. In M, we can only build with prime blocks. And all number structures will have a rectangular shape. Next, we are given

which is the 2-block in M, which we interpret as 2 in V. Since we cannot make a 3-block from the 2-block, we need a new prime block at the next stage, the 3-block.

How about that 4? That’s right. We can use the 2-blocks to make 4.

And this is how all of the number structures look in M. They are rectangular shapes with a prime base. The prime base of the 4-structure is the 2-block. Next, let us try to make 5. Since we can’t make a rectangular shape for 5 with either the 2-blocks or the 3-blocks, we will need to ask for a new prime block. At any stage, if we cannot make a rectangular shape for the number structure from previous prime blocks, we can ask for a new prime block. If we can actually make a number structure from previous prime blocks and we ask for a prime block, they will tell us to try again. Let’s ask for the 5-block now. Can we please have a new prime block?

Here it has arrived.

How can we make 6 from previous prime blocks? For 6, we can make a number structure from either the 2-blocks or the 3-blocks. In M, these are seen as different structures, but for us they are both have 6 individual blocks, so they are equal. We would think that any two number structures n and k are equal if |n| = |k|.

Here are the 6-structures.

For the 7-block we will need to request a new prime block. For 8, we can create from the 2-blocks. For 9, we can create from the 3-blocks. For 10, we can create from both the 2-blocks and the 5-blocks. For 11, we will need to ask for a new prime block, and so on.

Here are the structures for 1 through 11.

M IS CLOSED UNDER ADDITION AND MULTIPLICATION

You might be wondering, exactly who is constructing these number structures? They will get quite large and cumbersome. I’m happy to share that The Doozers (featured in the documentary “Fraggle Rock” by Jim Henson in the 1980s) are behind the scenes constructing all of the number structures. They are known for their superb engineering skills and strong work ethic. In order to add and multiply in M, we will need the help of the doozers so that when we add number structures in M we get another number structure in M.

Let us see the doozers at work to add the 3-block and the 5-block.

Step 1: Bust up the blocks/structures into individual squares.

Step 2: Try to build the result with prime blocks, starting with the 2-block.

In this case, we got lucky and were able to build the 8-structure with a 2-block base. If we can’t make the structure from 2-blocks, try a 3-block base, then try a 5-block base, etc. If we can’t make the structure from any prime blocks, then we will see that the sum is prime.

For example, let’s see the doozers construct 5 + 6. We can use either 6-structure.

We see again that the number structures have to be built out of prime blocks. When you connect a prime number amount of individual blocks, they stick together and light up, like in the pictures. When the doozers add in this way, the sum of any two number structures is another number structure in M. Try it for yourself. If you have some beads or dice, grab a random amount and try to build a rectangular structure from prime bases, starting with 2, then 3 and so on. Did you end up with a prime block? If not, what prime base worked?

To multiply two number structures in M, the doozers need to order more blocks to fulfill the order. In order to figure out how many prime blocks to order, they devised a scaffolding procedure where they use a temporary structure to measure how many prime blocks to order.

I’ll just let the doozers explain it. Here they are multiplying the 3-block by the 5-block.

Step 1: Choose one of the numbers’ prime base, and bust up the other number.

Here the doozers decided to use the 3-block from the first number as the prime base.

Step 2: Set up the busted up number as scaffolding to measure how many prime blocks to order.

For building structures, we can only use stacks of horizontal prime blocks. But we see now (and will see later) that vertical stacks do exist, like the scaffolding here. However, these vertical stacks are not stable enough to build with and can only be used for measuring (just like how we can’t build with stacks of the 1-block).

Step 3: Order more prime bases until it reaches the top of the scaffolding. Then remove the scaffolding.

If one of the numbers is composite, do the same process with each prime block that makes up the number structure - reusing the scaffolding for each prime block. Then stack everything on top of each other for the final result.

If one of the numbers is prime, we can always choose that as the prime base. If one of the numbers is a muted prime, we cannot use it as a prime block base, but we can use it to measure (we’ll see the muted primes very soon).

Let’s see the doozers multiply the 4-structure by the 9-structure.

KILLING PRIMES IN M

Legend says that the first prime was killed by a doozer with a broken heart. In order to forget about the number two as much as possible, the idea for a new world was created when the doozer kicked the 2-block vertically.

A new world was created where the rebellious doozer could work in peace and no longer have to build with the 2-block. Some of the other doozers wanted to go to this new world too, out of curiosity for most, but for some they liked the idea of not building with the 2-block. It was the least stable of all of the building blocks afterall.

No longer a prime block to be used in building structures, the 2-block was now a 2-stack. Still unique like our prime 2, but no longer creative in being able to be a prime base for a number structure. We will see that the 2-stack can still be used to add and multiply (as a measuring tool), but can't be used as a prime base.

What happens when 2 cannot be used as a prime base? Will another prime block appear? Looking at this extension through the eyes of V, everything will seem the same; we still have all of the natural numbers and the extension is still closed under addition and multiplication (and gives the correct result).

Let us see what is going on in an extension of M, a world called M⁻². Besides that the stable 2-block is now the unstable 2-stack, the main difference between M and the extension is that when we get to stage 4, we can no longer use the 2-blocks to build so we have to ask for a new prime block at this stage. Thus in M⁻², the number structure for 4 is a prime block, the 4-block.

We also see that in M⁻², the 6-structure, for example, has only one representation. And any n-structure with 2 in its prime factorization no longer has a representation with a 2-block prime base. However, there is much the same between M and M⁻², they have almost the same set of primes. Let P be the set of primes in V. A number structure is prime in M if it is a prime block. Then the set of primes in M, call it Pᴹ, is equal to P. And, the set of primes in M⁻², is equal to Pᴹ∖{2}∪{4}. In general, if we kill a prime p, then in M⁻ᵖ (the extension of M where p has been killed) the set of primes is Pᴹ∖{p}∪{p²}.

Theorem: If p is prime in M, then there is an extension M⁻ᵖ where p is no longer prime, and M⁻ᵖ ⊧ p² is prime. In addition, all other prime numbers are preserved in M⁻ᵖ, and there are no other new primes.

Proof: Suppose p is a prime block in M. Then, by construction, p is not prime in M⁻ᵖ since it has been killed. Since the only way to build the structure for p² in M is with the p-block, and there is no p-block in M⁻ᵖ, at stage p², we must request a new prime block. Thus p² is prime in M⁻ᵖ. Suppose q ≠ p is prime in M. Then at stage q in M⁻ᵖ, there are still no other prime blocks that can be used to make q if there weren’t any in M. And the only new prime is p², which is not a factor of q since q is really prime in V. If c not a prime block of M (hence composite in V), then c is a product of primes. If p is not a factor of c, then construction in M⁻ᵖ will be exactly as construction in M since the same prime blocks will work. If p is a factor of c, then since c is composite it has some other prime r its factorization. Thus, at stage c in M⁻ᵖ, use r-blocks to construct c. Thus c is not a prime block in M⁻ᵖ. ☐

Once M⁻² was open for business, soon all of the other M⁻ᵖ worlds opened up. What happened next? You guessed it. The doozers started experimenting with killing two primes at once. If we go to a world where p and q, distinct primes in M, are unable to create, what are the new prime blocks that are created? We know for sure that p² and q² are new prime blocks in the extension. Are there any other new primes? That’s right, the semiprime p⋅q will be a prime block in the extension where p and q are killed, since there is no other way to construct the p⋅q-structure so it becomes a p⋅q-block.

How many new primes are there in an extension of M where 3 distinct primes are killed? Let’s say the killed primes are p, q, and r. There’s 3 new primes from the squares of each of the 3 killed: { p², q², r² }. Then a new prime for every 2 of the 3, the semiprimes from the set of 3 primes: { p⋅q, p⋅r, q⋅r }. And then there’s a new prime which is the product of the 3 primes killed: { p⋅q⋅r }. So that’s 3 choose 3, plus 3 choose 2, plus 3 choose 1. Don’t you just love it when Pascal’s triangle shows up?

Therefore, in an extension of M, where we have killed n distinct primes of M, there will be 2ⁿ − 1 many new primes. And, since we killed n many primes in the process, there will be 2ⁿ − 1 − n many more primes in the extension than there were in M.

Theorem: If S is a set of n many primes in M, then in an extension of M where exactly the primes in S are killed, there will be 2ⁿ − 1 many new primes.

What will happen if we kill the prime p² in M⁻ᵖ?

What happens if we kill all of the infinitely many primes in M which are primes in V? Will there be any numbers left? Will all of the previously composite numbers become prime? Will it destroy the world? I don’t want to deprive you of the joy of finding out on your own. Imagine all of the primes that we know are not functional as building blocks, and then go through each stage and see when you need to request a new prime block. At any previously prime block stage, you will get a stack instead, but it can’t be used as a building block. See you soon…

Let’s start writing the prime blocks in M⁻ᴾ, the extension of M where all of the prime blocks of M are muted. The first prime block in M⁻ᴾ is the 4-block since there is no way to make 4 otherwise. The next prime block in M⁻ᴾ is 6 since we cannot make a 6-structure from the 4-block. The number 8 can be made with 4-blocks. For the number 9, since we cannot construct it from 4-blocks or 6-blocks, we have the 9-block in this extension. The number 12 can be made from 4-blocks or 6-blocks. The number 14 cannot be made from previous blocks, so 14 is prime in M⁻ᴾ.

Primes in M⁻ᴾ: {4, 6, 9, 10, 14, 15, 21, 22, 25, 26 . . . }. The new primes in M⁻ᴾ are the numbers of the form p², where p is prime in M, and all composites c in M whose prime factorization has prime powers only equal to one. Suppose m is some other composite in M whose prime factorization contains some q², where q is prime in M. Then since q was prime in M, it was killed in M⁻ᴾ, and so q² is prime in M⁻ᴾ. Then m can be constructed in M⁻ᴾ using q²-blocks.

Theorem: If P is the set of primes in M, then in the extension M⁻ᴾ where all of these primes are killed, there are still infinitely many prime numbers (and infinitely many composite numbers). In particular, the new primes are previous composites which are either of the form p², where p is prime in V, or have only powers of primes exactly one in their prime factorization.

It looks like in M⁻ᴾ, there are several primes which are right next to each other (like our 2 and 3), perhaps we can call them true twin primes. It looks like there are about as many twin prime pairs in M as there are true twin prime pairs, at least up to the primes we computed so far, in M⁻ᴾ. It is very fun to start to make observations about M⁻ᴾ, and to compare it to M. I don’t know about the relationship between the twin primes of M and the true twin primes of M⁻ᴾ, but let’s talk some more about twin primes and the twin prime conjecture next in the applications.

In M⁻ᴾ, the process of addition is exactly the same as in M, even if one or both of the summands is a muted prime since we only need to break into individual pieces and then build with the prime blocks of M⁻ᴾ. To multiply in M⁻ᴾ, if one of the factors is a prime block of M⁻ᴾ or a structure made from prime blocks, then do multiplication just as in M using prime blocks and scaffolding. If multiplying two muted primes in M⁻ᴾ, we know the result must be a prime block since any semiprimes from V are now prime, so the process is almost the same in that we can use the muted primes to measure how many individual blocks there are in the result (not thinking of either as a prime block when we lay down the stacks to measure), and then put all of those individual blocks together to give the result as a prime.

By the way, that original doozer, the one that kicked the 2-block, well he ended up liking his new world M⁻².

APPLICATIONS

The doozers love the prime blocks in their world because they love to build. Do you know what else the doozers love? Ever since the new worlds opened up, they love long-standing open problems about primes. They like that we don’t know the answer (though they would rejoice to hear about any new theorems about primes) so that we can make theorems about what if.

Once the doozers started getting into the theorem-making business, they decided to have an annual meeting to hear what everyone thinks about all of the worlds and their perspective from their respective worlds. Thankfully, I was able to sneak in with a press pass and I caught a few of the talks from the meeting. Here are some of the popular theorems.

Theorem: There is an extension of M where a prime number amount of primes from M are killed and the cardinality of the number of new primes is also prime.

Proof: Let p be a Mersenne prime. Then p = 2ⁿ − 1, where n is prime in M. Go to N, an extension of M, where n many primes have been killed. Then in N, there are 2ⁿ − 1 many new primes. Since 2ⁿ − 1 = p is prime, N has a prime number amount of new primes. Thus in N, a prime number n amount of primes from M were killed, and there are a prime number p amount of new primes. Further, if there are infinitely many Mersenne primes, then there are infinitely many such worlds. ☐

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This talk was about twin primes in an extension of M. The speaker was interested in killing twin primes in M, and getting a new twin pair in the extension N. She saw that if she killed a twin prime pair (p,q) of M in the extension, and one of {p² − 2, p² + 2, q² − 2, q² + 2} was prime in M, then in the extension there would be a new prime pair since both p² and q² are prime in the extension. She showed us this table of her findings for the first 13 twin primes in M:

She soon found out from the math stack exchange, that it is not known if there are infinitely many primes of the form of p² − 2, where p is prime, but there seems to be a lot.

Theorem: If there are infinitely many primes of the form of p² − 2, then there is an extension of M where the twin prime conjecture is true.

Proof: Let S = { p | p is prime in M and p² − 2 is prime in M}. Go to an extension of M by killing exactly the primes in S. Suppose N is the extension of M where of the primes in S are muted, then in N all of the elements of the set { p² | p ∈ S } are prime in N, and is infinite since S is infinite. And in N, since we did not kill any other primes, this set of primes { p² − 2 | p ∈ S } is still an infinite set of primes in N. Then, in N, the set of prime pairs { (p² − 2, p²) | p ∈ S } is infinite. ☐

Theorem: If there are infinitely many twin prime pairs (p, q) in M such that one of the pair (p² − 2, q² − 2) is prime in M, then there is an extension of M where infinitely many twin primes are killed, but for each one killed there is at least one more twin prime pair in the extension so that in the extension there are still infinitely many twin prime pairs.

Proof: Suppose there are infinitely many twin primes (p, q) in M, and that infinitely many of them have that at least one of the pair (p² − 2, q² − 2) is prime in M. Let S be the infinite set of such twin primes. Then go to an extension of M by killing exactly the primes in S. Let N be the extension of M where the twin primes of S have been killed. Then since we did not kill any other primes, all of the pairs (p² − 2, q² − 2) with (p, q) in S, still have at least one prime in N. Then since all of the squares of the primes that were killed are now prime in N, we have a new twin prime pair - either (p² −2, p) or (q² −2, q) in N, for every twin prime pair (p, q) that has been killed. ☐

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The last talk I was able to attend was about Goldbach’s conjecture. I had to step out before the questions at the end of the talk, because I could already feel it was getting heated. I could hear the debate from the hallway as I walked to the lobby for some peace and quiet, to think about what I had seen.

Theorem: If, in M, there is a very large even number e such that e is not the sum of two primes in M, then there is an extension N of M, where e is the sum of two primes in N.

Proof: Suppose that, in M, there is an even number e such that e cannot be written as the sum of two primes in M. By Chen’s theorem every sufficiently large even number can be written as a sum of two primes or the sum of a prime and a semiprime. Let us suppose that e is large enough so that Chen’s theorem holds. Since e is not the sum of two primes, it must be the sum of a prime and a semiprime. Suppose e = p + q ⋅ r, where p, q, and r are prime in M. Then go to an extension N of M by killing exactly primes q and r. Then, in N, the prime p is still prime since it has not been killed. And, since q and r were killed, the semiprime q ⋅ r is prime in N. Thus, in N, the even number e is the sum of two primes. ☐

The speaker noted that in the extension N, it is possible that even number q + r in N may not be able to be written as the sum of two primes in N since q and r are not prime in N, and it is possible that q + r cannot be written as the sum of any two other primes in M. So that in healing an exception to Goldbach’s conjecture you might create another exception. Some doozers argued that you cannot really kill an instance of the truth of Goldbach’s conjecture, because muted primes still count. What do you think? What kinds of theorems about extensions of M come to your mind?

As I was sitting in the lobby after the talks, the room was soon filled with doozers discussing the philosophy of their worlds.

Well, I bet you are ready to get back to V where everything is solid. Right? If you can make it through the doozers, you can get out the way you got in through the portal in the lobby. Or just stay forever!

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