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<div><span class="kw">theorem </span><a NAME="T21"><span class="comment"><font color="firebrick">:: PELLS_EQ:21</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">D</font> being   non  <a href="pythtrip.html#V1" title="PYTHTRIP:attr.1">square</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/> ( <font color="Olive" title="b2">p</font> is   <a href="pells_eq.html#V1" title="PELLS_EQ:attr.1">positive</a>   <a href="pells_eq.html#M1" title="PELLS_EQ:mode.1">Pell's_solution</a> of <font color="Olive" title="b1">D</font> iff  ex <font color="Olive" title="b3">n</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Olive" title="b1">D</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pells_eq.html#K1" title="PELLS_EQ:func.1">min_Pell's_solution_of</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pells_eq.html#K1" title="PELLS_EQ:func.1">min_Pell's_solution_of</a> <font color="Olive" title="b1">D</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K3" title="SQUARE_1:func.3">sqrt</a> <font color="Olive" title="b1">D</font></span>)</span></span>)</span></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b3">n</font> )</div></div>
