[Verse 1]
When we have a number a and a prime p too
There's a symbol that tells us what we need to do
Put a over p in parentheses neat
The Legendre symbol makes number theory complete
[Chorus]
One if it's a quadratic residue
Negative one if it's not, that's true
Zero when p divides a clean
The Legendre symbol shows what we mean
Q-R or not, the symbol will say
One, negative one, or zero today
[Verse 2]
A quadratic residue means there's an x
Where x squared equals a modulo p, no hex
If such a number x exists out there
The symbol equals one, beyond compare
[Chorus]
One if it's a quadratic residue
Negative one if it's not, that's true
Zero when p divides a clean
The Legendre symbol shows what we mean
Q-R or not, the symbol will say
One, negative one, or zero today
[Bridge]
When p divides a evenly
The symbol becomes zero, can't you see
No quadratic residue when divisible
The Legendre symbol stays predictable
[Verse 3]
If a is not a Q-R mod p
Then negative one is what we'll see
Three simple values, easy to recall
The Legendre symbol explains it all
[Chorus]
One if it's a quadratic residue
Negative one if it's not, that's true
Zero when p divides a clean
The Legendre symbol shows what we mean
Q-R or not, the symbol will say
One, negative one, or zero today
[Outro]
Legendre symbol, showing the way
In number theory every day
One, negative one, or zero
Our mathematical hero