[Verse 1] Count the primes from one to ten Two, three, five, and seven then Pi of x will tell us how Many primes we've counted now As numbers grow the primes spread out But they never fade or run out [Chorus] Pi of x over natural log of x That's the theorem we can't forget Primes thin out but never die Logarithmically they multiply One over ln x shows the way Probability of primes today [Verse 2] Take a number way up high What's the chance it's prime? We try One divided by its log Gives the odds through morning fog Million, billion, trillion more Primes get rare but there's no floor [Chorus] Pi of x over natural log of x That's the theorem we can't forget Primes thin out but never die Logarithmically they multiply One over ln x shows the way Probability of primes today [Bridge] As x approaches infinity The ratio's reality Pi of x and x over ln x Dance together, never vexed Asymptotic harmony Prime number symphony [Verse 3] Bertrand's postulate guarantees Between n and two n you'll see At least one prime will always live This pattern primes will always give Distribution has its laws to follow No empty gaps too large to swallow [Chorus] Pi of x over natural log of x That's the theorem we can't forget Primes thin out but never die Logarithmically they multiply One over ln x shows the way Probability of primes today [Outro] Count them up from small to large Primes are thinning but still at large Natural log holds the key To their density mystery
← Definition and Fundamental Theorem | Infinitude of Primes →