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<div><span class="kw">theorem </span><a NAME="T31"><span class="comment"><font color="firebrick">:: INT_8:31</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">p</font> being   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>  st <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp; <font color="Olive" title="b1">i</font>,<font color="Olive" title="b3">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  &amp; <font color="Olive" title="b1">i</font> <a href="int_8.html#R2" title="INT_8:pred.2">is_primitive_root_of</a> <font color="Olive" title="b3">p</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="pepin.html#K1" title="PEPIN:func.1">^2</a></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1</div></div>
