The divisor function σ_k(n) sums the k-th powers of all divisors of n. Its 2-adic valuation — how many times 2 divides the result — is bounded above by a function of n alone: roughly log₂ n for odd k, and a similar expression for even k. These bounds come from the arithmetic of carrying in binary, the way power sums interact with prime factorization. They are continuous in spirit: as n grows, the bound grows logarithmically, and most integers fall well below it.
Cheng and Zhang (arXiv:2603.11979) prove these bounds are tight and characterize when equality holds. For odd k, the result is startling: equality occurs if and only if n is a product of distinct Mersenne primes — primes of the form 2^p − 1.
Mersenne primes are famous for their own reasons. They are the subject of the longest-running computational search in mathematics. Whether infinitely many exist is unknown. They appear in the theory of perfect numbers (every even perfect number has the form 2^(p−1)(2^p − 1) where 2^p − 1 is prime), in coding theory, and in the structure of finite simple groups. They were not invited into 2-adic valuation theory.
Yet the bound converges to them. The algebraic reason is traceable: a Mersenne prime q = 2^p − 1 has the property that σ_k(q) = 1 + q^k, and q ≡ −1 mod 2^p, so the 2-adic valuation of 1 + q^k is exactly p when k is odd. A product of distinct Mersenne primes multiplies these contributions without interference. No other factorization structure achieves this: composite numbers leak valuation at the wrong places, and non-Mersenne primes don't carry cleanly enough.
For even k, the answer is more extreme: equality holds only when n = 3. The landscape collapses to a single point.
The pattern: an analytic inequality, derived from the internal mechanics of a number-theoretic function, optimized to its boundary, selects objects that were already independently remarkable. The bound did not know about Mersenne primes. It was not constructed to detect them. It was constructed to measure carrying behavior in divisor sums, and at the extremal edge of that measurement, Mersenne primes are what carrying behavior looks like when it works perfectly.
This is distinct from cases where a known structure is used to prove a bound (the way E8 is used to prove sphere-packing optimality in dimension 8). Here the structure emerges from the bound. The question “which integers maximize the 2-adic content of their divisor sums?” was not a question about Mersenne primes. It became one.
The phenomenon might be called convergent selection: independent mathematical structures, developed for unrelated purposes, converge on the same objects when pushed to extremes. The objects are not remarkable because they appear in multiple contexts — they appear in multiple contexts because the algebraic properties that make them Mersenne primes are the same properties that saturate the 2-adic carrying bound. The question is whether we should be surprised. Possibly not. The surprise is that we are.