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<i><font color="Green">E44</font></i>: 257 <a href="binarith.html#K3">-'</a> 1 = 
257 <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/>.= 
256 <a href="nat_1.html#K1">+</a> 0

;<br/>
<i><font color="Green">E45</font></i>: 2 <a href="series_1.html#K3">to_power</a> 8 = 
2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">4 <a href="nat_1.html#K1">+</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
16 <a href="nat_1.html#K2">*</a> 16
<b>by </b><i>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
256

;<br/>
<i><font color="Green">E49</font></i>: 8 <a href="binarith.html#K3">-'</a> 1 = 
8 <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/>.= 
7 <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>256 <a href="nat_1.html#K5">div</a> 2 = 
2 <a href="series_1.html#K3">to_power</a> <span class="p1">(<span class="default">8 <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, <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">4 <a href="nat_1.html#K1">+</a> 3</span>)</span>
<b>by </b><i>, <a class="ref" href="power.html#T46">POWER:46</a></i>
<br/>.= 
16 <a href="nat_1.html#K2">*</a> 8
<b>by </b><i>, , <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
128

;<br/>
<a NAME="E6:117"/><b>then </b>
 <a href="pepin.html#K5" target="_self">Fermat</a> 3 <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> 3</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">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>, , , </i>;<br/>
<a NAME="E7:117"/><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> 3</span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">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> 3
 
<b>by </b><i><a class="ref" href="int_2.html#T19">INT_2:19</a></i>;<br/>
<b>hence </b><a NAME="E8:117"/>
257 is <a href="int_2.html#V1">prime</a>
 <b>by </b><i>, </i>;<br/>


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