The nth prime number can be approximated by a formula that depends not on logarithms or sieves but on the divisor function — the count of how many ways each preceding integer can be factored. Botkin, Dawsey, Hemmer, Just, and Schneider show that p_n is approximately 1 plus twice the sum of the ceiling of d(j)/2 for j from 1 to n-1, where d(j) counts the divisors of j. The error term is negligible. The formula recovers the prime number theorem as a corollary and — more unexpectedly — produces estimates of the prime-counting function that are more accurate than the logarithmic integral for values up to 10,000.

The surprise is the direction of the connection. Partition theory — the study of how integers decompose into sums — and multiplicative number theory — the study of how integers decompose into products — have historically been treated as separate branches with occasional formal analogies. This formula links them mechanically. The additive structure of partitions, as encoded in the divisor function, directly constrains where the primes must fall. The primes are not distributed independently of the divisor landscape; they are positioned by it.

The model also predicts the twin prime conjecture as a natural consequence rather than an independent hypothesis. If the relationship between divisor sums and prime positions holds asymptotically, then the gaps between consecutive primes cannot avoid being small infinitely often.

The through-claim is architectural. When two mathematical domains appear independent, finding a formula that converts one into the other suggests the independence was an artifact of the tools, not of the mathematics. The primes were always encoded in the divisor function. The encoding just needed to be read in the right direction — from additive decomposition toward multiplicative structure, rather than the reverse.