[Verse 1]
When numbers dance in modular space
And primes define the playing field
Some integers find their perfect match
A square root hiding what they yield
Take twenty-five in mod thirteen
Does some x squared equal this scene?
[Chorus]
Quadratic residues, they're squares in disguise
Hidden beneath modular skies
When x squared congruent a mod p appears
That's when the quadratic residue clears
Euler's criterion shows the way
Raise to power (p minus one) half today
[Verse 2]
If the result is positive one
Then a quadratic residue you've won
But if negative one comes to light
No square root exists in sight
The exponent holds the secret key
(p minus one) divided by two, you see
[Chorus]
Quadratic residues, they're squares in disguise
Hidden beneath modular skies
When x squared congruent a mod p appears
That's when the quadratic residue clears
Euler's criterion shows the way
Raise to power (p minus one) half today
[Bridge]
Prime p must be odd and strange
For this theorem to arrange
Legendre symbols tell the tale
Plus one or minus one, without fail
[Verse 3]
In crypto systems they appear
Making some computations clear
While others become quite hard to solve
This asymmetry helps problems revolve
From ancient Greeks to modern codes
Quadratic residues share their loads
[Chorus]
Quadratic residues, they're squares in disguise
Hidden beneath modular skies
When x squared congruent a mod p appears
That's when the quadratic residue clears
Euler's criterion shows the way
Raise to power (p minus one) half today
[Outro]
So when you see a mod p equation
Test for quadratic demonstration
Euler's test will never deceive
The answer that you will receive