[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
← Euler's Theorem (Generalization) | The Law of Quadratic Reciprocity →