{"id":"palomar/PALOMAR-2026-08-27-000006","url":"https://github.com/AviKndr/universal-words/tree/90e3cbf743db257ecc13639f024c52fb197d6a48/lean","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-08-27-000006-v1.json","repo":"https://github.com/AviKndr/universal-words","tags":["math.GR","math.NT","msc:20B30","msc:20D60","msc:11P32","msc:05A05","group-theory","number-theory","word-maps","symmetric-groups"],"score":0,"title":"AviKndr/universal-words","trust":"high","domain":"math.GR","source":"palomar","authors":["Avi Kondareddy"],"reviews":0,"subject":"math.GR","summary":"Answer to Problem 10.32 of the Kourovka Notebook, posed by Yu. I. Merzlyakov in 1986 and still open in the Notebook's 21st edition (2026), in both directions. Positive direction: there are constants C and N0 such that for all nonzero integers r, s with r or s odd, the word x^r y^s is universal on the symmetric group S_n (every permutation is a value) whenever n >= N0 and n >= C * log m(r,s), where m(r,s) is the product of the distinct primes dividing rs. This is the S_n analogue of the logarithmic Brenner-Evans-Silberger bound for A_n; the best previous S_n bound (Droste, 1986) was linear in…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-08-27","reactions":{},"formalized":true}