document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=475364>Question 475364</A>: <I><SPAN STYLE=\"word-break: break-all;\"> let m and n be two consicutive even integers and 1/m+1/n=p/q (in the lowest form) prove that (p, q, q+1) is a pythagorous triplet. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #326405 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=326405\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Instead of writing n we can write m+2, in which\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{1}{m} + \frac{1}{m+2} = \frac{p}{q} \Rightarrow \frac{2m + 2}{m^2 + 2m} = \frac{p}{q}\">\r<BR>\n" );
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document.write( "And we want to prove that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p^2 + q^2 = (q+1)^2 = q^2 + 2q + 1 \Rightarrow p^2 = 2q + 1\">\r<BR>\n" );
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document.write( "Begin by letting m = 2k, so our equation becomes\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{4k+2}{4k^2 + 4k} = \frac{p}{q}\">\r<BR>\n" );
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document.write( "Simplify the LHS by dividing both top and bottom by 2:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{2k+1}{2k^2 + 2k} = \frac{p}{q}\">\r<BR>\n" );
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document.write( "The denominator is equal to 2k(k+1), in which k+1, 2k, 2k+1 are all pairwise relatively prime. Hence, the LHS is irreducible, so we can say that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p = 2k + 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE q = 2k^2 + 2k\">\r<BR>\n" );
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document.write( "And that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE p^2 = 2q + 1 = 4k^2 + 4k + 1\">, so we are done.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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