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

<a NAME="E1:46"/><i><font color="Green">E45</font></i>: 
4001 <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="binop_2.html#K24">*</a> 5</span>)</span> <a href="binop_2.html#K24">*</a> 5</span>)</span> <a href="binop_2.html#K24">*</a> 32
 
;<br/>
<i><font color="Green">E46</font></i>: <span class="p1">(<span class="default">5 <a href="binop_2.html#K24">*</a> 5</span>)</span> <a href="binop_2.html#K24">*</a> 5 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">5 <a href="wsierp_1.html#K2">|^</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> 5</span>)</span> <a href="binop_2.html#K24">*</a> 5
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<span class="p1">(<span class="default">5 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> 5
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
5 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
5 <a href="wsierp_1.html#K2">|^</a> 3

;<br/>
<i><font color="Green">E47</font></i>: 5 <a href="nat_1.html#K8">hcf</a> 32 = 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">5 <a href="binop_2.html#K24">*</a> 6</span>)</span></span>)</span> <a href="nat_1.html#K8">hcf</a> 5

<br/>.= 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K8">hcf</a> 2
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
1 <a href="nat_1.html#K8">hcf</a> 2
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="newton.html#T64">NEWTON:64</a></i>
;<br/>
<a NAME="E4:46"/>
2 <a href="wsierp_1.html#K2">|^</a> 1 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<a NAME="E5:46"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 1</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>;<br/>
<i><font color="Green">E49</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 2</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E5:46"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
4
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<i><font color="Green">E50</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 4</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
<span class="p1">(<span class="default">4 <a href="binop_2.html#K24">*</a> 4</span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E6:46"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
16
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<i><font color="Green">E51</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 8</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 4</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
<span class="p1">(<span class="default">16 <a href="binop_2.html#K24">*</a> 16</span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E7:46"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
256
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<a NAME="E9:46"/>
256 <a href="binop_2.html#K24">*</a> 256 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 16</span>)</span> <a href="binop_2.html#K23">+</a> 1520
 
;<br/>
<a NAME="E10:46"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default">256 <a href="binop_2.html#K24">*</a> 256</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 <a href="hidden.html#R1">=</a> 1520
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E53</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 16</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 8</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
1520
<b>by </b><i><a class="txt" href="nat_4.html#E8:46"><i><font color="Green">E51</font></i></a>, <a class="txt" href="nat_4.html#E10:46"><i><font color="Green">E52</font></i></a>, <a class="ref" href="nat_4.html#T11" target="_self">Th11</a></i>
;<br/>
<a NAME="E12:46"/>
1520 <a href="binop_2.html#K24">*</a> 1520 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 577</span>)</span> <a href="binop_2.html#K23">+</a> 1823
 
;<br/>
<a NAME="E13:46"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default">1520 <a href="binop_2.html#K24">*</a> 1520</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 <a href="hidden.html#R1">=</a> 1823
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E55</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 32</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 16</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
1823
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E11:46"><i><font color="Green">E53</font></i></a>, <a class="txt" href="nat_4.html#E13:46"><i><font color="Green">E54</font></i></a></i>
;<br/>
<a NAME="E15:46"/>
1823 <a href="binop_2.html#K24">*</a> 1823 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 830</span>)</span> <a href="binop_2.html#K23">+</a> 2499
 
;<br/>
<a NAME="E16:46"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default">1823 <a href="binop_2.html#K24">*</a> 1823</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 <a href="hidden.html#R1">=</a> 2499
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E57</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 64</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 32</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
2499
<b>by </b><i><a class="txt" href="nat_4.html#E14:46"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E16:46"><i><font color="Green">E56</font></i></a></i>
;<br/>
<a NAME="E18:46"/>
2499 <a href="binop_2.html#K24">*</a> 2499 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 1560</span>)</span> <a href="binop_2.html#K23">+</a> 3441
 
;<br/>
<a NAME="E19:46"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default">2499 <a href="binop_2.html#K24">*</a> 2499</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 <a href="hidden.html#R1">=</a> 3441
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E59</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 128</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 64</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
3441
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E17:46"><i><font color="Green">E57</font></i></a>, <a class="txt" href="nat_4.html#E19:46"><i><font color="Green">E58</font></i></a></i>
;<br/>
<a NAME="E21:46"/>
3441 <a href="binop_2.html#K24">*</a> 3441 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 2959</span>)</span> <a href="binop_2.html#K23">+</a> 1522
 
;<br/>
<a NAME="E22:46"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default">3441 <a href="binop_2.html#K24">*</a> 3441</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 <a href="hidden.html#R1">=</a> 1522
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E61</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 256</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 128</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
1522
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E20:46"><i><font color="Green">E59</font></i></a>, <a class="txt" href="nat_4.html#E22:46"><i><font color="Green">E60</font></i></a></i>
;<br/>
<a NAME="E24:46"/>
1522 <a href="binop_2.html#K24">*</a> 1522 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 578</span>)</span> <a href="binop_2.html#K23">+</a> 3906
 
;<br/>
<a NAME="E25:46"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default">1522 <a href="binop_2.html#K24">*</a> 1522</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 <a href="hidden.html#R1">=</a> 3906
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E63</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 512</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 256</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
3906
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E23:46"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nat_4.html#E25:46"><i><font color="Green">E62</font></i></a></i>
;<br/>
<a NAME="E27:46"/>
3906 <a href="binop_2.html#K24">*</a> 3906 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 3813</span>)</span> <a href="binop_2.html#K23">+</a> 1023
 
;<br/>
<a NAME="E28:46"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default">3906 <a href="binop_2.html#K24">*</a> 3906</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 <a href="hidden.html#R1">=</a> 1023
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E65</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 1024</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 512</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
1023
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E26:46"><i><font color="Green">E63</font></i></a>, <a class="txt" href="nat_4.html#E28:46"><i><font color="Green">E64</font></i></a></i>
;<br/>
<a NAME="E30:46"/>
1023 <a href="binop_2.html#K24">*</a> 1023 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 261</span>)</span> <a href="binop_2.html#K23">+</a> 2268
 
;<br/>
<a NAME="E31:46"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default">1023 <a href="binop_2.html#K24">*</a> 1023</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 <a href="hidden.html#R1">=</a> 2268
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E67</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 2048</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 1024</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
2268
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E29:46"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nat_4.html#E31:46"><i><font color="Green">E66</font></i></a></i>
;<br/>
<a NAME="E33:46"/><i><font color="Green">E68</font></i>: 
3441 <a href="binop_2.html#K24">*</a> 1823 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 1567</span>)</span> <a href="binop_2.html#K23">+</a> 3376
 
;<br/>
<a NAME="E34:46"/><i><font color="Green">E69</font></i>: 
1522 <a href="binop_2.html#K24">*</a> 3376 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 1284</span>)</span> <a href="binop_2.html#K23">+</a> 988
 
;<br/>
<a NAME="E35:46"/><i><font color="Green">E70</font></i>: 
3906 <a href="binop_2.html#K24">*</a> 988 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 964</span>)</span> <a href="binop_2.html#K23">+</a> 2164
 
;<br/>
<a NAME="E36:46"/><i><font color="Green">E71</font></i>: 
1023 <a href="binop_2.html#K24">*</a> 2164 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 553</span>)</span> <a href="binop_2.html#K23">+</a> 1219
 
;<br/>
<a NAME="E37:46"/><i><font color="Green">E72</font></i>: 
2268 <a href="binop_2.html#K24">*</a> 1219 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 691</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
;<br/>
<i><font color="Green">E73</font></i>: 4001 <a href="binarith.html#K5">-'</a> 1 = 
4001 <a href="binop_2.html#K10">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
4000

;<br/>
<i><font color="Green">E74</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">4001 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2048 <a href="binop_2.html#K23">+</a> 1952</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E38:46"><i><font color="Green">E73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 2048</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 1952</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 2048</span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default">1024 <a href="binop_2.html#K23">+</a> 928</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 1024</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 928</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E32:46"><i><font color="Green">E67</font></i></a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default">512 <a href="binop_2.html#K23">+</a> 416</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E29:46"><i><font color="Green">E65</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 512</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 416</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">3906 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">256 <a href="binop_2.html#K23">+</a> 160</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E26:46"><i><font color="Green">E63</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">3906 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 256</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 160</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">3906 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">1522 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default">128 <a href="binop_2.html#K23">+</a> 32</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E23:46"><i><font color="Green">E61</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">3906 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">1522 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 128</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 32</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">3906 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">1522 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">3441 <a href="binop_2.html#K24">*</a> 1823</span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E14:46"><i><font color="Green">E55</font></i></a>, <a class="txt" href="nat_4.html#E20:46"><i><font color="Green">E59</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">3906 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">1522 <a href="binop_2.html#K24">*</a> 3376</span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E33:46"><i><font color="Green">E68</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">3906 <a href="binop_2.html#K24">*</a> 988</span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E34:46"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1023 <a href="binop_2.html#K24">*</a> 2164</span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E35:46"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">2268 <a href="binop_2.html#K24">*</a> 1219</span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E36:46"><i><font color="Green">E71</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="nat_4.html#E37:46"><i><font color="Green">E72</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
;<br/>
<a NAME="E40:46"/><i><font color="Green">E75</font></i>: 
4000 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">5 <a href="binop_2.html#K24">*</a> 800</span>)</span> <a href="binop_2.html#K23">+</a> 0
 
;<br/>
<a NAME="E41:46"/>
2 <a href="wsierp_1.html#K2">|^</a> 800 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="prepower.html#T19">PREPOWER:19</a></i>;<br/>
<a NAME="E42:46"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 800</span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 800</span>)</span> <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E43:46"/><i><font color="Green">E77</font></i>: 
1522 <a href="binop_2.html#K24">*</a> 1823 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 693</span>)</span> <a href="binop_2.html#K23">+</a> 1913
 
;<br/>
<a NAME="E44:46"/><i><font color="Green">E78</font></i>: 
3906 <a href="binop_2.html#K24">*</a> 1913 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 1867</span>)</span> <a href="binop_2.html#K23">+</a> 2311
 
;<br/>
<i><font color="Green">E79</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 800</span>)</span> <a href="nat_1.html#K4">mod</a> 4001 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">512 <a href="binop_2.html#K23">+</a> 288</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 512</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 288</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">3906 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default">256 <a href="binop_2.html#K23">+</a> 32</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E26:46"><i><font color="Green">E63</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">3906 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 256</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 32</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">3906 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1522 <a href="binop_2.html#K24">*</a> 1823</span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E23:46"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nat_4.html#E14:46"><i><font color="Green">E55</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">3906 <a href="binop_2.html#K24">*</a> 1913</span>)</span> <a href="nat_1.html#K4">mod</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E43:46"><i><font color="Green">E77</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
<br/>.= 
2311
<b>by </b><i><a class="txt" href="nat_4.html#E44:46"><i><font color="Green">E78</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
;<br/>
<a NAME="E46:46"/><i><font color="Green">E80</font></i>: 
6311 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 2310
 
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">4001 <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> 5</span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K8">hcf</a> 4001 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 800</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K8">hcf</a> 4001
<b>by </b><i><a class="txt" href="nat_4.html#E38:46"><i><font color="Green">E73</font></i></a>, <a class="txt" href="nat_4.html#E40:46"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 800</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">4001 <a href="binop_2.html#K24">*</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K8">hcf</a> 4001
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
4001 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 800</span>)</span> <a href="binop_2.html#K23">+</a> 4000</span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E42:46"><i><font color="Green">E76</font></i></a>, <a class="ref" href="scmp_gcd.html#T1">SCMP_GCD:1</a></i>
<br/>.= 
4001 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2311 <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default">4000 <a href="nat_1.html#K4">mod</a> 4001</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E45:46"><i><font color="Green">E79</font></i></a>, <a class="ref" href="euler_2.html#T8">EULER_2:8</a></i>
<br/>.= 
4001 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2311 <a href="binop_2.html#K23">+</a> 4000</span>)</span> <a href="nat_1.html#K4">mod</a> 4001</span>)</span>
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2310 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1691</span>)</span> <a href="nat_1.html#K8">hcf</a> 2310
<b>by </b><i><a class="txt" href="nat_4.html#E46:46"><i><font color="Green">E80</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
<br/>.= 
1691 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1691 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 619</span>)</span>
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">619 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 453</span>)</span> <a href="nat_1.html#K8">hcf</a> 619
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">453 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 166</span>)</span> <a href="nat_1.html#K8">hcf</a> 453
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
166 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">166 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 121</span>)</span>
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">121 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 45</span>)</span> <a href="nat_1.html#K8">hcf</a> 121
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
45 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">45 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 31</span>)</span>
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">31 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 14</span>)</span> <a href="nat_1.html#K8">hcf</a> 31
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
14 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">14 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 3</span>)</span>
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">3 <a href="binop_2.html#K24">*</a> 4</span>)</span> <a href="binop_2.html#K23">+</a> 2</span>)</span> <a href="nat_1.html#K8">hcf</a> 3
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
2 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
2 <a href="nat_1.html#K8">hcf</a> 1
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="newton.html#T64">NEWTON:64</a></i>
;<br/>
<b>hence </b><a NAME="E48:46"/>
4001 is <a href="int_2.html#V1">prime</a>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:46"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nat_4.html#E2:46"><i><font color="Green">E46</font></i></a>, <a class="txt" href="nat_4.html#E3:46"><i><font color="Green">E47</font></i></a>, <a class="ref" href="pepin.html#T64">PEPIN:64</a>, <a class="txt" href="nat_4.html#E39:46"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_4.html#T25" target="_self">Th25</a></i>;<br/>


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