The Lewis Williams who served in the 14th (McCarver's) Arkansas Infantry and later the 21st Arkansas Infantry is a different man. He was born in 1816 in Mecklenburg County, Virginia, and was discharged for being over-age.

There were other men named Lewis Williams who are also unrelated to your Lewis Williams. They served in 10th Arkansas Infantry, 13th Arkansas Infantry, 37th Arkansas Infantry, and 58th Arkansas Militia. Also a Lewis B. Williams (11th Arkansas Infantry) and a Lewis H. Williams (9th Arkansas), both unrelated to your research.

Lewis seems to have a pretty popular name in the Williams clan.