Universality thresholds of power-product words in symmetric groups

For nonzero integers r,s with at least one odd, when is every permutation in Sₙ expressible as xʳyˢ? Let m(r,s) be the product of distinct primes dividing rs; the quantitative target is logarithmic dependence on m.

Why it matters and what counts as progress

Why it matters: A sharp threshold scale improves earlier linear bounds for universality of this word map.

What counts as progress: A proof for all sufficiently large n≥C log m(r,s), and examples requiring logarithmic order, with the odd-exponent assumption retained.

How the problem developed

1986 · Open problem [1]
Yu. I. Merzlyakov Kourovka Notebook Problem 10.32 asked when the word x^r y^s is universal on S_n, including the case where at least one exponent is odd.

AI contribution

Claude-led proof gives a logarithmic symmetric-group word threshold

For nonzero integers r,s with at least one odd, the word xʳyˢ is universal in Sₙ once n is sufficiently large and at least C log m(r,s), where m is the product of the distinct primes dividing rs. A separate construction shows the logarithmic order cannot generally be reduced.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4 kernel with Mathlib, axiom audit and Comparator)

Human understandingFew