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Algebraic independence of logarithms of distinct primes

About 16 years old · traced to

Let m≥1m\geq 1 and let p1,…,pmp_1,\ldots,p_m be distinct primes. The field Q(z1,…,zm)\mathbb Q(z_1,\ldots,z_m) of rational functions in z1,…,zmz_1,\ldots,z_m is evaluated at zj=ln⁡pjz_j=\ln p_j by the homomorphism

ψ:Q(z1,…,zm)⟶Q(ln⁡p1,…,ln⁡pm).\psi:\mathbb Q(z_1,\ldots,z_m)\longrightarrow\mathbb Q(\ln p_1,\ldots,\ln p_m).

Algebraic-independence conjecture. The numbers ln⁡p1,…,ln⁡pm\ln p_1,\ldots,\ln p_m are algebraically independent. Equivalently, ψ\psi is an isomorphism.

The conjecture is presented as a consequence of Schanuel's conjecture: unique factorization gives rational independence of the logarithms, and Schanuel's conjecture would then imply the asserted transcendence degree. Its unconditional status is not established in the supplied text.

References

Primary source

Wayne M Lawton, “Joint distribution of leftmost digits in positional notation and Schanuels's conjecture”, arXiv:2603.03110 (2026).

Additional references

7 papers in this index state this conjecture (2010–2026). The statement above is taken from the most recent of them; the others are arXiv:2512.14077, arXiv:2512.03432, arXiv:1807.11044, arXiv:1603.05155, arXiv:1407.5165, arXiv:1011.3368.

algebraic-independencelogarithmstranscendence-theory
Posted · edited

Progress summary

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Open

The conjecture that logarithms of different primes have no hidden polynomial relations remains unproved, with only a conditional route through Schanuel's conjecture.

The conjecture asserts algebraic independence of ln⁡p1,…,ln⁡pm\ln p_1,\ldots,\ln p_m for distinct primes. The supplied literature presents it as a consequence of Schanuel's conjecture, but reports no unconditional proof or disproof.

Known results

Schanuel's conjecture would imply that Q\mathbb{Q}-linearly independent logarithms of algebraic numbers are algebraically independent; distinct-prime logarithms satisfy the required rational independence. The relevant logarithmic special case is described as equivalent to Roy's Structural Rank Conjecture, not proved.

Current status (as of October 2026): The conjecture remains open; no unconditional result, counterexample, claimed resolution, or verification was found.

Sources

Solutions 0

No solutions have been posted yet.