<i>&gt; I assume the author means that the point of contradiction doesn&#x27;t rest on the &quot;divisibility properties&quot; of the integer</i><p>I see what the author meant by that. I just think it was stated in a slightly exaggerated way. The &quot;divisibility properties&quot; of the integers are still used. That part has just has been moved to another corner of the proof, by transforming fractional equations.<p>One of the comments in the article is quite interesting here:<p><i>| Theodor Estermann proved the irrationality of 2√ without relying on the prime factorization of m.</i><p>I believe that this statement is more correct. The &quot;traditional&quot; proof uses not just divisibility, but prime factorization, which is quite a strong property. And that is something the alternative proof doesn&#x27;t make use of.<p>Maybe the introduction should have been stated that way.