document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208295>Question 1208295</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If 2¹³ + 2¹⁰ + 2ˣ = y², <BR>\n" );
document.write( "find x and y.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846623 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=846623\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll assume x and y are nonnegative integers.<BR>\n" );
document.write( "There might be a more clever and elegant solution, but I'm going to check possible values of x.\r<BR>\n" );
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document.write( "If x = 0, then,<BR>\n" );
document.write( "2^13 + 2^10 + 2^x <BR>\n" );
document.write( "= 2^13 + 2^10 + 2^0<BR>\n" );
document.write( "= 9217<BR>\n" );
document.write( "And,<BR>\n" );
document.write( "sqrt(9217) = 96.0052 approximately<BR>\n" );
document.write( "Since the square root result isn't a whole number, this shows 9217 isn't a perfect square.<BR>\n" );
document.write( "y^2 = 9217 doesn't have an integer solution.<BR>\n" );
document.write( "x = 0 won't pair with an integer y value.<BR>\n" );
document.write( "We can rule out x = 0.\r<BR>\n" );
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document.write( "Let's try x = 1.<BR>\n" );
document.write( "2^13 + 2^10 + 2^x <BR>\n" );
document.write( "= 2^13 + 2^10 + 2^1<BR>\n" );
document.write( "= 9218<BR>\n" );
document.write( "And,<BR>\n" );
document.write( "sqrt(9218) = 96.0104 approximately<BR>\n" );
document.write( "Same idea as the previous paragraph. <BR>\n" );
document.write( "We can rule out x = 1.\r<BR>\n" );
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document.write( "Let's try x = 2.<BR>\n" );
document.write( "2^13 + 2^10 + 2^x <BR>\n" );
document.write( "= 2^13 + 2^10 + 2^2<BR>\n" );
document.write( "= 9220<BR>\n" );
document.write( "And,<BR>\n" );
document.write( "sqrt(9220) = 96.0208 approximately<BR>\n" );
document.write( "We can rule out x = 2 for similar reasoning as the previous paragraphs.\r<BR>\n" );
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document.write( "And so on.<BR>\n" );
document.write( "Keep this process going until reaching x = 14.<BR>\n" );
document.write( "Use of a spreadsheet is strongly recommended to make this process very quick.<BR>\n" );
document.write( "Alternatively you can use a programming language such as Python to write up a quick script.<BR>\n" );
document.write( "2^13 + 2^10 + 2^x <BR>\n" );
document.write( "= 2^13 + 2^10 + 2^14<BR>\n" );
document.write( "= 25600<BR>\n" );
document.write( "And,<BR>\n" );
document.write( "sqrt(25600) = 160 exactly<BR>\n" );
document.write( "We finally get an integer result. <BR>\n" );
document.write( "This proves that <font color=red>x = 14 and y = 160</font> is one ordered pair solution.<BR>\n" );
document.write( "2^13 + 2^10 + 2^<font color=red>14</font> = <font color=red>160</font>^2 \r<BR>\n" );
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document.write( "There might be other solutions.<BR>\n" );
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