Euclid’s proof of the infinity of primes, in-verse
Over numbers and their combinations if you sit and mull
You will find that not one of them is uninteresting and dull.
But it is a certain class of figures that most attention stirs
Yes, I am speaking of those special ones, the prime numbers.
Prime numbers are interesting, the mathematician posits,
‘Cos they make up all the others, the so-called composites.
Here’s an imperfect analogy, a simple little working rule,
Consider the prime to be an atom, then a composite’s a molecule.
To carry the chemical analogy completely out of bound,
Consider these atoms (primes) as randomly strewn around
Some here, others there, their patterns concealed
Few sequences stand out, no deep design is revealed.
As the ladder of digits you will climb
Hoping to predict the appearance of a prime
One fact stands out, above all else, as you stare
Prime numbers become more and more rare.
In other words, there are fewer primes the higher you go
Does this heightened rarity mean something, or no?
Is there a prime that is the biggest one?
If there is, finding it could be fun.
Of course, there’s another option, it is clear
Is it that primes get rarer but never disappear?
They get fewer and fewer the further we see
But they never really get done, on to infinity.
Which of these two options is the one that’s true
And how much arithmetic do you have to do
To show whether the primes are finite or not?
Ideally via a nice elegant proof, in one clean shot.
Euclid, the geometry guy, comes to the rescue
And produced a cool little proof, whew!
So here’s my attempt to show in rhyme
There’s no such thing as the largest prime.
But don’t take my word for this information
Let us take a moment to look at Euclid’s creation.
Now here’s a critical (and smart) mathematical move
Euclid said, that in order the converse to prove,
Let’s start by assuming there is one (whatever it may be)
Let’s give it a name, this biggest prime, let’s call it p.
This largest prime, (p) when all is said and done
Is divisible just by itself and the number 1.
Now lets multiply together all these primes we have, then
We get a humungous number, we shall, for now, call N.
(This number we shall capitalize
To represent its rather large numerical size.)
Because N is the product of every prime we know,
It is divisible by every one of them, that’s easy to show.
Now take this N and to it, just add 1 (that’s all)
A difference that you may consider as insignificant and small.
But think about it for a moment and you may see
What a difference this addition makes to divisibility.
This new number N + 1 you will soon realize
In the case of divisibility can really surprise.
Take any number on our list including p, our largest prime.
N+1 is not divisible by any of these atoms of the number line.
Whichever way you put N+1 through the division blender
You will always be left behind with one solitary remainder!
Stay with us for a moment, we are almost done
In fact, this is where it gets to be kinda fun.
Notice, there are just two possibilities at this juncture
Let us, in turn, consider each conjecture.
Either N + 1 is a prime, or it is not.
If it is a prime, our assumption is shot!
For N + 1 is clearly
Much bigger than p!
Something that contradicts what we started with
The idea that p is the largest prime must be a myth
Or, it could be that N+1 is a number composite
Implying there are some primes that can cleanly divide it.
One thing for sure, this divisor cannot be
A prime in our original list we see.
‘cos we just showed that dividing N+1
By primes in the first list just cannot be done!
This just means, there are some primes we missed
When we were building our initial complete list.
And if we missed one, you can sure
There are just an infinity more.
Isn’t it time you said to me
Those magical words, QED.
To sum up, the finite prime set idea is pure fiction
Since assuming it leads to a contradiction.
Primes may be rarer and rarer the higher we go
But they do go on forever, and this Euclid did show.