[Verse 1]
Two primes standing all alone
P and Q, each on their own
But there's a secret they both share
A golden bond beyond compare
When P asks Q "am I a square?"
The answer's linked through ancient prayer
[Chorus]
It's quadratic reciprocity
The golden theorem's mystery
P over Q times Q over P
Equals one or negative
Check if both are three mod four
Then it's minus one for sure
Otherwise it's always one
Gauss's golden theorem won
[Verse 2]
Take seventeen and twenty-three
Independent as can be
But ask if seventeen's a square
Modulo twenty-three with care
The answer tells you something more
About twenty-three's square lore
[Chorus]
It's quadratic reciprocity
The golden theorem's mystery
P over Q times Q over P
Equals one or negative
Check if both are three mod four
Then it's minus one for sure
Otherwise it's always one
Gauss's golden theorem won
[Bridge]
Minus one over P depends
On P mod four, the rule extends
If P is one mod four, it's plus
If P is three mod four, no fuss, it's minus
Two over P has its own way
P squared minus one divided by eight holds sway
[Verse 3]
Primes don't know about each other
Yet they dance like sister, brother
This conspiracy runs deep
Ancient secrets that they keep
From quadratic reciprocity
To Langlands' grand symphony
[Chorus]
It's quadratic reciprocity
The golden theorem's mystery
P over Q times Q over P
Equals one or negative
Check if both are three mod four
Then it's minus one for sure
Otherwise it's always one
Gauss's golden theorem won
[Outro]
Six proofs Gauss gave us all
For this theorem we recall
The beginning of the tale
Where deeper laws will never fail