[Verse 1]
Let me tell you 'bout a question old
Are there primes beyond what we've been told
Euclid had a clever way to see
That prime numbers stretch to infinity
He said suppose we list them all complete
Every prime from two up to the greatest feat
Multiply them all and add just one
Watch what happens when this trick is done
[Chorus]
Infinitely many primes exist
No matter how long you make your list
Euclid's proof and Euler's sum both show
The primes go on wherever numbers go
One over p will always grow
The primes go on wherever numbers go
[Verse 2]
Take that number N we just created
When we divide by primes we've calculated
The remainder's always one you'll find
No prime on our list can break divide
Either N itself must be a prime
We missed it in our listing time
Or N has factors we don't know
New primes that help the number grow
[Chorus]
Infinitely many primes exist
No matter how long you make your list
Euclid's proof and Euler's sum both show
The primes go on wherever numbers go
One over p will always grow
The primes go on wherever numbers go
[Bridge]
Euler found another way to prove
Sum of one over p will never move
To a finite bound like squares would do
The reciprocals diverge right through
This tells us more than just they're endless
The primes are dense enough and boundless
Their reciprocal sum grows without end
On this mathematical fact depend
[Verse 3]
So when you think you've found them all
The primes will answer back the call
Contradiction shows our assumption's wrong
The sequence of primes goes on and on
[Final Chorus]
Infinitely many primes exist
No finite bound can hold this list
Euclid's logic, Euler's sum both show
The primes go on wherever numbers go
Mathematics proves what we need to know
The primes go on wherever numbers go
[Outro]
From two to three to five and on
The prime parade keeps marching strong