[Verse 1]
Numbers have their secrets hiding in plain sight
Two and three and five stand alone in mathematical light
They're divisible by one and themselves, nothing more
These are primes, the building blocks at math's core
Seven, eleven, thirteen march in endless line
Each one atomic, pure, by definition's design
[Chorus]
Every number breaks apart like molecules to atoms
Primes are the foundation that mathematics builds its patterns
Unique factorization, no exceptions to the rule
Two to the third times five squared, this theorem is our tool
P-one to A-one, P-two to A-two
Every integer greater than one, this formula rings true
[Verse 2]
Start with strong induction, prove existence first
If the number's prime already, then our thirst is quenched
If composite, split it into factors A and B
Both are smaller, both have prime factorizations, you see
Multiply those factorizations, now we've shown
Every number has a breakdown it can call its own
[Chorus]
Every number breaks apart like molecules to atoms
Primes are the foundation that mathematics builds its patterns
Unique factorization, no exceptions to the rule
Two to the third times five squared, this theorem is our tool
P-one to A-one, P-two to A-two
Every integer greater than one, this formula rings true
[Bridge]
But uniqueness needs more proof than just existence
Euclid's lemma cuts through all resistance
If prime P divides A times B, then it's clear
P divides A or P divides B, that's what we revere
Through Bézout's magic, this truth we can see
The factorization's unique as unique can be
[Verse 3]
In some number systems, this breaks down completely
Z-bracket-root-negative-five fails so neatly
Six equals two times three, but also other ways
Algebraic number theory studies where uniqueness frays
But integers are special, their atomic structure's sound
The most fundamental theorem that in math can be found
[Outro]
Primes are atoms, composites are molecules
This theorem forms the base of all our mathematical jewels