Define A(n) as the set of integers a less than n where two conditions hold simultaneously: n divides a² - 1, and a divides n² - 1. Each condition alone admits many solutions — the number of a satisfying n | a² - 1 grows with the divisor count of n, and similarly for the reverse. Together, they collapse the solution space to almost nothing.
The paper proves |A(n)| < log₂(n), with an asymptotic average just above 2. Computations up to 10⁷ suggest |A(n)| ≤ 3, though this stronger bound remains unproven.
The mechanism connecting the two conditions runs through Fibonacci-like polynomials. The mutual divisibility constraint forces a into algebraic relations that are vastly more restrictive than either condition independently. What looks like a symmetric doubling of constraints acts as a multiplicative stranglehold — two moderate restrictions, mirrored, produce an almost-empty set.
This is not unique to this problem. Reciprocal constraints in number theory are generically devastating. When n constrains a and a simultaneously constrains n, the mutual dependence eliminates degrees of freedom faster than additive counting suggests. The solutions that survive are not “twice as constrained” — they are the tiny residue that satisfies a relationship tighter than either factor.
The gap between the proven logarithmic bound and the empirical bound of 3 is itself instructive. The true answer may be that reciprocal divisibility is even more annihilating than the proof captures — that the log₂(n) bound is an artifact of proof technique rather than the structure of the problem.
Symmetry in constraints does not double the difficulty. It squares it. And the square of a moderate restriction is often indistinguishable from a prohibition.