Adibelli, Azem BerivanGoral, Haydar2025-08-272025-08-2720250002-98901930-0972https://doi.org/10.1080/00029890.2025.2525049https://hdl.handle.net/11147/18378We first prove the infinitude of the primes via a special case of Rado's theorem whose proof is based on the infinite Ramsey theorem. In the proof, we use the colorings of the positive integers introduced by Levent Alpoge [1] and Andrew Granville [2]. Finally, using Rado's theorem for integral domains, we will give another proof for the infinitude of nonassociated prime elements in any unique factorization domain R with a few units.eninfo:eu-repo/semantics/closedAccessArticle2-s2.0-10501252163410.1080/00029890.2025.2525049