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

<a NAME="E1:45"/><i><font color="Green">E44</font></i>: 
2503 <a href="binop_2.html#K10">-</a> 1 <a href="hidden.html#R1">=</a> 139 <a href="binop_2.html#K24">*</a> 18
 
;<br/>
<a NAME="E2:45"/><i><font color="Green">E45</font></i>: 
139 <a href="hidden.html#R1">=</a> 139 <a href="wsierp_1.html#K2">|^</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<i><font color="Green">E46</font></i>: 139 <a href="nat_1.html#K8">hcf</a> 18 = 
<span class="p1">(<span class="default">13 <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">18 <a href="binop_2.html#K24">*</a> 7</span>)</span></span>)</span> <a href="nat_1.html#K8">hcf</a> 18

<br/>.= 
13 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">13 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 5</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">3 <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> 5</span>)</span></span>)</span> <a href="nat_1.html#K8">hcf</a> 5
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
3 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">3 <a href="binop_2.html#K24">*</a> 1</span>)</span></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">1 <a href="binop_2.html#K23">+</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#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="E4:45"/>
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="E5:45"/><b>then </b><i><font color="Green">E47</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> 2503 <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">E48</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> 2503 = 
<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> 2503

<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> 2503
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E5:45"><i><font color="Green">E47</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">E49</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> 2503 = 
<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> 2503

<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> 2503
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E6:45"><i><font color="Green">E48</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">E50</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> 2503 = 
<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> 2503

<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> 2503
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E7:45"><i><font color="Green">E49</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="E9:45"/>
256 <a href="binop_2.html#K24">*</a> 256 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 26</span>)</span> <a href="binop_2.html#K23">+</a> 458
 
;<br/>
<a NAME="E10:45"/><b>then </b><i><font color="Green">E51</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> 2503 <a href="hidden.html#R1">=</a> 458
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E52</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> 2503 = 
<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> 2503

<br/>.= 
458
<b>by </b><i><a class="txt" href="nat_4.html#E8:45"><i><font color="Green">E50</font></i></a>, <a class="txt" href="nat_4.html#E10:45"><i><font color="Green">E51</font></i></a>, <a class="ref" href="nat_4.html#T11" target="_self">Th11</a></i>
;<br/>
<a NAME="E12:45"/>
458 <a href="binop_2.html#K24">*</a> 458 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 83</span>)</span> <a href="binop_2.html#K23">+</a> 2015
 
;<br/>
<a NAME="E13:45"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default">458 <a href="binop_2.html#K24">*</a> 458</span>)</span> <a href="nat_1.html#K4">mod</a> 2503 <a href="hidden.html#R1">=</a> 2015
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E54</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> 2503 = 
<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> 2503

<br/>.= 
2015
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E11:45"><i><font color="Green">E52</font></i></a>, <a class="txt" href="nat_4.html#E13:45"><i><font color="Green">E53</font></i></a></i>
;<br/>
<a NAME="E15:45"/>
2015 <a href="binop_2.html#K24">*</a> 2015 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 1622</span>)</span> <a href="binop_2.html#K23">+</a> 359
 
;<br/>
<a NAME="E16:45"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default">2015 <a href="binop_2.html#K24">*</a> 2015</span>)</span> <a href="nat_1.html#K4">mod</a> 2503 <a href="hidden.html#R1">=</a> 359
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E56</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> 2503 = 
<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> 2503

<br/>.= 
359
<b>by </b><i><a class="txt" href="nat_4.html#E14:45"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E16:45"><i><font color="Green">E55</font></i></a></i>
;<br/>
<a NAME="E18:45"/>
359 <a href="binop_2.html#K24">*</a> 359 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 51</span>)</span> <a href="binop_2.html#K23">+</a> 1228
 
;<br/>
<a NAME="E19:45"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default">359 <a href="binop_2.html#K24">*</a> 359</span>)</span> <a href="nat_1.html#K4">mod</a> 2503 <a href="hidden.html#R1">=</a> 1228
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E58</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> 2503 = 
<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> 2503

<br/>.= 
1228
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E17:45"><i><font color="Green">E56</font></i></a>, <a class="txt" href="nat_4.html#E19:45"><i><font color="Green">E57</font></i></a></i>
;<br/>
<a NAME="E21:45"/>
1228 <a href="binop_2.html#K24">*</a> 1228 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 602</span>)</span> <a href="binop_2.html#K23">+</a> 1178
 
;<br/>
<a NAME="E22:45"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default">1228 <a href="binop_2.html#K24">*</a> 1228</span>)</span> <a href="nat_1.html#K4">mod</a> 2503 <a href="hidden.html#R1">=</a> 1178
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E60</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> 2503 = 
<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> 2503

<br/>.= 
1178
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E20:45"><i><font color="Green">E58</font></i></a>, <a class="txt" href="nat_4.html#E22:45"><i><font color="Green">E59</font></i></a></i>
;<br/>
<a NAME="E24:45"/>
1178 <a href="binop_2.html#K24">*</a> 1178 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 554</span>)</span> <a href="binop_2.html#K23">+</a> 1022
 
;<br/>
<a NAME="E25:45"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default">1178 <a href="binop_2.html#K24">*</a> 1178</span>)</span> <a href="nat_1.html#K4">mod</a> 2503 <a href="hidden.html#R1">=</a> 1022
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E62</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> 2503 = 
<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> 2503

<br/>.= 
1022
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E23:45"><i><font color="Green">E60</font></i></a>, <a class="txt" href="nat_4.html#E25:45"><i><font color="Green">E61</font></i></a></i>
;<br/>
<a NAME="E27:45"/>
1022 <a href="binop_2.html#K24">*</a> 1022 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 417</span>)</span> <a href="binop_2.html#K23">+</a> 733
 
;<br/>
<a NAME="E28:45"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default">1022 <a href="binop_2.html#K24">*</a> 1022</span>)</span> <a href="nat_1.html#K4">mod</a> 2503 <a href="hidden.html#R1">=</a> 733
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E64</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> 2503 = 
<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> 2503

<br/>.= 
733
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E26:45"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nat_4.html#E28:45"><i><font color="Green">E63</font></i></a></i>
;<br/>
<a NAME="E30:45"/>
733 <a href="binop_2.html#K24">*</a> 733 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 214</span>)</span> <a href="binop_2.html#K23">+</a> 1647
 
;<br/>
<a NAME="E31:45"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default">733 <a href="binop_2.html#K24">*</a> 733</span>)</span> <a href="nat_1.html#K4">mod</a> 2503 <a href="hidden.html#R1">=</a> 1647
 
<b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<i><font color="Green">E66</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> 2503 = 
<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> 2503

<br/>.= 
1647
<b>by </b><i><a class="ref" href="nat_4.html#T11" target="_self">Th11</a>, <a class="txt" href="nat_4.html#E29:45"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nat_4.html#E31:45"><i><font color="Green">E65</font></i></a></i>
;<br/>
<a NAME="E33:45"/><i><font color="Green">E67</font></i>: 
359 <a href="binop_2.html#K24">*</a> 64 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 9</span>)</span> <a href="binop_2.html#K23">+</a> 449
 
;<br/>
<a NAME="E34:45"/><i><font color="Green">E68</font></i>: 
1228 <a href="binop_2.html#K24">*</a> 449 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 220</span>)</span> <a href="binop_2.html#K23">+</a> 712
 
;<br/>
<a NAME="E35:45"/><i><font color="Green">E69</font></i>: 
1178 <a href="binop_2.html#K24">*</a> 712 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 335</span>)</span> <a href="binop_2.html#K23">+</a> 231
 
;<br/>
<a NAME="E36:45"/><i><font color="Green">E70</font></i>: 
1647 <a href="binop_2.html#K24">*</a> 231 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 152</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
;<br/>
<i><font color="Green">E71</font></i>: 2503 <a href="binarith.html#K5">-'</a> 1 = 
2503 <a href="binop_2.html#K10">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
2502

;<br/>
<i><font color="Green">E72</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2503 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503 = 
<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> 454</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E37:45"><i><font color="Green">E71</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> 454</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<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> 2503</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">256 <a href="binop_2.html#K23">+</a> 198</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <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> 198</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E32:45"><i><font color="Green">E66</font></i></a>, <a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <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">128 <a href="binop_2.html#K23">+</a> 70</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E23:45"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <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> 128</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 70</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1228 <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">64 <a href="binop_2.html#K23">+</a> 6</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E20:45"><i><font color="Green">E58</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1228 <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> 64</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 6</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1228 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">359 <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">4 <a href="binop_2.html#K23">+</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E17:45"><i><font color="Green">E56</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1228 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">359 <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> 4</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1228 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">359 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">16 <a href="binop_2.html#K24">*</a> 4</span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E6:45"><i><font color="Green">E48</font></i></a>, <a class="txt" href="nat_4.html#E7:45"><i><font color="Green">E49</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1228 <a href="binop_2.html#K24">*</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default">359 <a href="binop_2.html#K24">*</a> 64</span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <a href="binop_2.html#K24">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">1228 <a href="binop_2.html#K24">*</a> 449</span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E33:45"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default">1647 <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1178 <a href="binop_2.html#K24">*</a> 712</span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E34:45"><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">1647 <a href="binop_2.html#K24">*</a> 231</span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E35:45"><i><font color="Green">E69</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#E36:45"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
;<br/>
<a NAME="E39:45"/><i><font color="Green">E73</font></i>: 
2502 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">139 <a href="binop_2.html#K24">*</a> 18</span>)</span> <a href="binop_2.html#K23">+</a> 0
 
;<br/>
<a NAME="E40:45"/>
2 <a href="wsierp_1.html#K2">|^</a> 18 <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="E41:45"/><b>then </b><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 18</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> 18</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/>
<i><font color="Green">E75</font></i>: <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 18</span>)</span> <a href="nat_1.html#K4">mod</a> 2503 = 
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">16 <a href="binop_2.html#K23">+</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 16</span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default">458 <a href="binop_2.html#K24">*</a> 4</span>)</span> <a href="nat_1.html#K4">mod</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E11:45"><i><font color="Green">E52</font></i></a>, <a class="txt" href="nat_4.html#E6:45"><i><font color="Green">E48</font></i></a>, <a class="ref" href="euler_2.html#T11">EULER_2:11</a></i>
<br/>.= 
1832
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<a NAME="E43:45"/><i><font color="Green">E76</font></i>: 
4334 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1831
 
;<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">2503 <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> 139</span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K8">hcf</a> 2503 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> 18</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K8">hcf</a> 2503
<b>by </b><i><a class="txt" href="nat_4.html#E37:45"><i><font color="Green">E71</font></i></a>, <a class="txt" href="nat_4.html#E39:45"><i><font color="Green">E73</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> 18</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> <span class="p2">(<span class="default">2503 <a href="binop_2.html#K24">*</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K8">hcf</a> 2503
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
2503 <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> 18</span>)</span> <a href="binop_2.html#K23">+</a> 2502</span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E41:45"><i><font color="Green">E74</font></i></a>, <a class="ref" href="scmp_gcd.html#T1">SCMP_GCD:1</a></i>
<br/>.= 
2503 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1832 <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default">2502 <a href="nat_1.html#K4">mod</a> 2503</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2503</span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E42:45"><i><font color="Green">E75</font></i></a>, <a class="ref" href="euler_2.html#T8">EULER_2:8</a></i>
<br/>.= 
2503 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1832 <a href="binop_2.html#K23">+</a> 2502</span>)</span> <a href="nat_1.html#K4">mod</a> 2503</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">1831 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 672</span>)</span> <a href="nat_1.html#K8">hcf</a> 1831
<b>by </b><i><a class="txt" href="nat_4.html#E43:45"><i><font color="Green">E76</font></i></a>, <a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>
<br/>.= 
672 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">672 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 487</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">487 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 185</span>)</span> <a href="nat_1.html#K8">hcf</a> 487
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
185 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">185 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 117</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">117 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 68</span>)</span> <a href="nat_1.html#K8">hcf</a> 117
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
68 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">68 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 49</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">49 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 19</span>)</span> <a href="nat_1.html#K8">hcf</a> 49
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
19 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">19 <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K23">+</a> 11</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">11 <a href="binop_2.html#K24">*</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 8</span>)</span> <a href="nat_1.html#K8">hcf</a> 11
<b>by </b><i><a class="ref" href="euler_1.html#T9">EULER_1:9</a></i>
<br/>.= 
8 <a href="nat_1.html#K8">hcf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">8 <a href="binop_2.html#K24">*</a> 1</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> 2</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="E45:45"/>
2503 is <a href="int_2.html#V1">prime</a>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:45"><i><font color="Green">E44</font></i></a>, <a class="txt" href="nat_4.html#E2:45"><i><font color="Green">E45</font></i></a>, <a class="txt" href="nat_4.html#E3:45"><i><font color="Green">E46</font></i></a>, <a class="ref" href="nat_4.html#T34" target="_self">Th34</a>, <a class="txt" href="nat_4.html#E38:45"><i><font color="Green">E72</font></i></a>, <a class="ref" href="nat_4.html#T25" target="_self">Th25</a></i>;<br/>


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