document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=691044>Question 691044</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a²+b²=25<BR>\n" );
document.write( "a³+b³=91 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #426840 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=426840\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I am not sure if the problem is meant as a system of non-linear equations, or a simpler problem, looking for integers that satisfy each (or both) equations.<BR>\n" );
document.write( " <BR>\n" );
document.write( "The most popular of Pythagorean triples (and the only one I remember) is<BR>\n" );
document.write( "(3,4,5) with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2%2B4%5E2=5%5E2 ALIGN=MIDDLE  ALT=\"3%5E2%2B4%5E2=5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2%2B4%5E2=25 ALIGN=MIDDLE  ALT=\"3%5E2%2B4%5E2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So the only answer to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2=25 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > with whole numbers is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%283%5E2%2B4%5E2=5%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%283%5E2%2B4%5E2=5%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If we allow negative numbers, there are other integer solutions to just that equation (we just cahnge the signs for a and/or b).<BR>\n" );
document.write( " <BR>\n" );
document.write( "Curiously, the same numbers 3 and 4 are a solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E3%2Bb%5E3=91 ALIGN=MIDDLE  ALT=\"a%5E3%2Bb%5E3=91\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E3%2B4%5E3=27%2B64=91 ALIGN=MIDDLE  ALT=\"3%5E3%2B4%5E3=27%2B64=91\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So you could say that the pair (3,4) is the only integer solution to the system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%5E2%2Bb%5E2=25%2Ca%5E3%2Bb%5E3=91%29 ALIGN=MIDDLE  ALT=\"system%28a%5E2%2Bb%5E2=25%2Ca%5E3%2Bb%5E3=91%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "If we wanted all real numbers that are solutions to the system, we would have to look for two more points where the graphs intersect for<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=25 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > (circle with radius 5) and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2By%5E3=91 ALIGN=MIDDLE  ALT=\"x%5E3%2By%5E3=91\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=root%283%2C91-x%5E3%29 ALIGN=MIDDLE  ALT=\"y=root%283%2C91-x%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (a snakey downwards line). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );