We illustrate how elementary information-theoretic ideas may be employed to provide proofs for well-known, nontrivial results in number theory. Specifically, we give an elementary and fairly short proof of the following asymptotic result, p≤n log p p ∼ log n, as n → ∞, where the sum is over all primes p not exceeding n. We also give finite-n bounds refining the above limit. This result, originally proved by Chebyshev in 1852, is closely related to the celebrated prime number theorem.