<?xml version="1.0"?>
<div class="add">

<i><font color="Green">E99</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default">256 <a href="nat_1.html#K2">*</a> 256</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">256 <a href="nat_1.html#K2">*</a> 256</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default">256 <a href="nat_1.html#K2">*</a> 256</span>)</span> <a href="nat_1.html#K1">+</a> 0

;<br/>
<i><font color="Green">E100</font></i>: 2 <a href="series_1.html#K3">to_power</a> 16 = 
2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">8 <a href="nat_1.html#K1">+</a> 8</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
256 <a href="nat_1.html#K2">*</a> 256
<b>by </b><i><a class="txt" href="pepin.html#E24"><i><font color="Green">Lemma20</font></i></a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
;<br/>
<i><font color="Green">E101</font></i>: 16 <a href="binarith.html#K5">-'</a> 1 = 
16 <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
15 <a href="nat_1.html#K1">+</a> 0

;<br/>
2 <a href="series_1.html#K3">to_power</a> 1 = 
2 <a href="wsierp_1.html#K2">|^</a> 1
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
2
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
;<br/>
<b>then </b><span class="p1">(<span class="default">256 <a href="nat_1.html#K2">*</a> 256</span>)</span> <a href="nat_1.html#K3">div</a> 2 = 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">16 <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pepin.html#E2:118"><i><font color="Green">E100</font></i></a>, <a class="ref" href="nat_2.html#T18">NAT_2:18</a></i>
<br/>.= 
2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">8 <a href="nat_1.html#K1">+</a> 7</span>)</span>
<b>by </b><i><a class="txt" href="pepin.html#E3:118"><i><font color="Green">E101</font></i></a>, <a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
256 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">4 <a href="nat_1.html#K1">+</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pepin.html#E24"><i><font color="Green">Lemma20</font></i></a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
256 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default">16 <a href="nat_1.html#K2">*</a> 8</span>)</span>
<b>by </b><i><a class="txt" href="pepin.html#E22"><i><font color="Green">Lemma18</font></i></a>, <a class="txt" href="pepin.html#E23"><i><font color="Green">Lemma19</font></i></a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
256 <a href="nat_1.html#K2">*</a> 128

;<br/>
<a NAME="E6:118"/><b>then </b>
 <a href="pepin.html#K5" target="_self">Fermat</a> 4 <a href="int_1.html#R2">divides</a> <span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 4</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="pepin.html#E1:118"><i><font color="Green">E99</font></i></a>, <a class="txt" href="pepin.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="txt" href="pepin.html#E121"><i><font color="Green">Lemma98</font></i></a>, <a class="ref" href="pepin.html#T59" target="_self">Th59</a></i>;<br/>
<a NAME="E7:118"/><b>then </b>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> 4</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4">-</a> 1 <a href="int_1.html#R1">are_congruent_mod</a>  <a href="pepin.html#K5" target="_self">Fermat</a> 4
 
<b>by </b><i><a class="ref" href="int_2.html#T19">INT_2:19</a></i>;<br/>
<b>hence </b><a NAME="E8:118"/>
<span class="p1">(<span class="default">256 <a href="nat_1.html#K2">*</a> 256</span>)</span> <a href="nat_1.html#K1">+</a> 1 is <a href="int_2.html#V1">prime</a>
 <b>by </b><i><a class="ref" href="pepin.html#T59" target="_self">Th59</a>, <a class="ref" href="pepin.html#T63" target="_self">Th63</a></i>;<br/>


</div>
