[Verse 1] Two primes meet at a party tonight They don't know they've been connected p and q in the dancing light Their histories intersected Is p a square when we mod by q? Is q a square when we mod by p? They hold the answers, it's strange but true In their prime identity [Chorus] It's the secret handshake, the hidden code Quadratic reciprocity shows What p knows about q, q knows about p It's symmetric, can't you see? The secret handshake of the primes They've been dancing all this time Legendre symbols tell the tale When both are one mod four, it never fails [Verse 2] But wait, there's a twist in this prime romance When both are three mod four The signs will flip, they'll change their dance Negative one times negative one makes the score Gauss discovered this hidden truth Structure where chaos seemed to reign The mathematical fountain of youth In number theory's grand domain [Chorus] It's the secret handshake, the hidden code Quadratic reciprocity shows What p knows about q, q knows about p It's symmetric, can't you see? The secret handshake of the primes They've been dancing all this time Legendre symbols tell the tale With sign correction, it will never fail [Bridge] Like strangers finding wedding photos Where they both appear in the crowd The primes reveal their secret memos Speaking truths both clear and proud From ancient Greece to Gauss's mind This theorem breaks the random wall Shows the patterns we can find In the greatest mystery of all [Chorus] It's the secret handshake, the hidden code Quadratic reciprocity shows What p knows about q, q knows about p It's symmetric, can't you see? The secret handshake of the primes They've been dancing all this time Check the residues, watch them shine In this mathematical design [Outro] So when two primes meet face to face They already know the other's grace The secret handshake, tried and true Quadratic reciprocity's gift to you
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