[Verse 1] Euclid had a clever scheme, assumed the list complete All primes from two to some large number, nice and neat But multiply them all and add just one to the mix This new number breaks the rules with mathematical tricks [Chorus] Primes go on forever, stretching past the stars No final destination, no matter how far Take the whole collection, multiply plus one Either prime itself or factors we've undone Infinitude of primes, the theorem rings true Mathematics proves what ancient Greeks once knew [Verse 2] Every prime divides our product, leaves remainder zero But our special number N makes each remainder hero One divided by each prime, when remainders stay Means no listed prime can factor N away [Chorus] Primes go on forever, stretching past the stars No final destination, no matter how far Take the whole collection, multiply plus one Either prime itself or factors we've undone Infinitude of primes, the theorem rings true Mathematics proves what ancient Greeks once knew [Bridge] Euler found another path through analysis refined Sum of prime reciprocals blows the finite mind One over p diverges, grows beyond all bounds Not just infinite in count, but dense throughout the rounds [Verse 3] If the primes were truly finite, sums would have a cap But the harmonic series shows there's still a gap Reciprocals keep climbing, never reach a ceiling Proving prime abundance with analytic feeling [Outro] From geometry to calculus, two proofs align Primes extend eternally in numerical design No last prime waiting somewhere at the cosmic edge Mathematics guarantees this ironclad pledge
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