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<b>let </b><font color="Maroon" title="c1">p</font> be   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>;<br/>

<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E33">A1</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 &amp; <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/>

<a NAME="E2:46"/><b>then </b><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 4</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 4</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E3:46"/>
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<a NAME="E4:46"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>;<br/>
<a NAME="E5:46"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 4</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="int_5.html#E2:46"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 4</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> 2</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T14" title="NEWTON:th.14">NEWTON:14</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2</span>)</span> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="wsierp_1.html#T2" title="WSIERP_1:th.2">WSIERP_1:2</a></i>
<br/>.= 
1 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 4</span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T100" title="NEWTON:th.100">NEWTON:100</a>, <a class="ref" href="square_1.html#T59" title="SQUARE_1:th.59">SQUARE_1:59</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="newton.html#T15" title="NEWTON:th.15">NEWTON:15</a></i>
;<br/>
<a NAME="E7:46"/><b>then </b>
 <a href="int_5.html#K2" title="INT_5:func.2">Lege</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="int_5.html#E1:46"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="int_5.html#T36" target="_self" title="INT_5:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E8:46"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_5.html#R1" title="INT_5:pred.1">is_quadratic_residue_mod</a> <font color="Maroon" title="c1">p</font>
 <b>by </b><i><a class="ref" href="int_5.html#D3" target="_self" title="INT_5:def.3">Def3</a></i>;<br/>


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