The Digit Conjecture

Benford’s law describes the distribution of leading digits in a single number base. Lawton asks: what happens when you look at leading digits simultaneously across multiple bases?

For two bases, the answer is clean. The joint distribution of leftmost digits in bases b₁ and b₂ covers all possible digit combinations if and only if log b₁ and log b₂ are rationally independent — they don’t share a common rational multiple. This is equivalent to saying the bases aren’t powers of a common integer.

For three or more bases, the “if” direction still holds. But the converse — proving that covering all combinations requires rational independence — needs a conjecture from transcendental number theory. Specifically, it requires that the logarithms of distinct primes are algebraically independent, a consequence of Schanuel’s conjecture.

The through-claim: a concrete question about digit patterns connects to one of the deepest open problems in number theory. Whether you can observe every possible combination of leading digits across three bases depends on whether certain numbers have hidden algebraic relationships. The digit question is accessible; the algebraic question is decades unsolved.

The connection runs through the equidistribution of logarithmic images on tori — the joint digit distribution is uniform when certain projections onto multidimensional tori are dense. Density depends on algebraic independence. The combinatorial surface conceals a transcendence problem.

Digits ask a question that number theory cannot yet fully answer.