[Verse 1]
Numbers scatter like stars across the endless line
Two, three, five, seven - building blocks divine
But as we climb the ladder, something strange occurs
The gaps between them widen, the pattern blurs
Pi of x counts the treasures hiding in the void
Prime counting function, never to be destroyed
[Chorus]
Pi of x is roughly x divided by natural log of x
As numbers grow enormous, this approximation rocks
One over ln of x tells the probability tale
That random integers near x as primes will never fail
Thinning logarithmically but never fade away
The Prime Number Theorem shows us how they play
[Verse 2]
Gauss gazed at tables, noticed something profound
The density decreases but primes can still be found
Near a thousand, one in seven might be prime
Near a million, one in fourteen marks the time
The logarithm governs this celestial dance
Mathematics gives us more than just a glance
[Chorus]
Pi of x is roughly x divided by natural log of x
As numbers grow enormous, this approximation rocks
One over ln of x tells the probability tale
That random integers near x as primes will never fail
Thinning logarithmically but never fade away
The Prime Number Theorem shows us how they play
[Bridge]
Infinity stretches before us, primes will never cease
Though separated further, they maintain their peace
The natural log increases, but oh so very slow
Ensuring prime abundance wherever we may go
[Chorus]
Pi of x is roughly x divided by natural log of x
As numbers grow enormous, this approximation rocks
One over ln of x tells the probability tale
That random integers near x as primes will never fail
Thinning logarithmically but never fade away
The Prime Number Theorem shows us how they play
[Outro]
From Euclid's ancient wisdom to modern computation
Primes distribute forever across the vast equation