https://number.subwiki.org/w/index.php?title=Quadratic_nonresidue&feed=atom&action=history Quadratic nonresidue - Revision history 2022-01-18T22:33:11Z Revision history for this page on the wiki MediaWiki 1.24alpha https://number.subwiki.org/w/index.php?title=Quadratic_nonresidue&diff=532&oldid=prev Vipul: Created page with '==Definition== Suppose $n$ is a natural number. A '''quadratic nonresidue''' modulo $n$ is a number $a$ (or a residue class of a number <mat…' 2010-05-29T21:58:02Z <p>Created page with &#039;==Definition== Suppose &lt;math&gt;n&lt;/math&gt; is a <a href="/w/index.php?title=Natural_number&amp;action=edit&amp;redlink=1" class="new" title="Natural number (page does not exist)">natural number</a>. A &#039;&#039;&#039;quadratic nonresidue&#039;&#039;&#039; modulo &lt;math&gt;n&lt;/math&gt; is a number &lt;math&gt;a&lt;/math&gt; (or a residue class of a number &lt;mat…&#039;</p> <p><b>New page</b></p><div>==Definition==<br /> <br /> Suppose &lt;math&gt;n&lt;/math&gt; is a [[natural number]]. A '''quadratic nonresidue''' modulo &lt;math&gt;n&lt;/math&gt; is a number &lt;math&gt;a&lt;/math&gt; (or a residue class of a number &lt;math&gt;a&lt;/math&gt;) relatively prime to &lt;math&gt;n&lt;/math&gt; such that the equation:<br /> <br /> &lt;math&gt;x^2 \equiv a \pmod n&lt;/math&gt;<br /> <br /> has no solution. Since &lt;math&gt;a&lt;/math&gt; is relatively prime to &lt;math&gt;n&lt;/math&gt;, it suffices to check that there is no solution with &lt;math&gt;x&lt;/math&gt; relatively prime to &lt;math&gt;n&lt;/math&gt;. Further, it suffices to check that there is no solution for &lt;math&gt;x&lt;/math&gt; relatively prime to &lt;math&gt;n&lt;/math&gt; and &lt;math&gt;0 \le x \le n&lt;/math&gt;.<br /> <br /> Note that the term ''quadratic nonresidue'' is used both for actual numbers and for residue classes. The term is ''not'' used in cases where &lt;math&gt;a&lt;/math&gt; is not relatively prime to &lt;math&gt;n&lt;/math&gt;.<br /> <br /> The opposite of quadratic nonresidue is [[quadratic residue]].</div> Vipul