Algebra.Com's Answer #846625 by ikleyn(52777)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "If 2¹³ + 2¹⁰ + 2ˣ = y², find x and y.
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document.write( " It is assumed that x and y are integer numbers.\r
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document.write( " I will give another, strict mathematical solution.\r
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document.write( " +  = 9216 = 96^2. (1)\r\n" );
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document.write( "Therefore,  =  -  . (2)\r\n" );
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document.write( "Hence, = (y+96)*(y-96). (3)\r\n" );
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document.write( "Thus, is the product of integers y+96 and y-96.\r\n" );
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document.write( "From it, we conclude that y+96 and y-96 are degrees of 2.\r\n" );
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document.write( "So, we write\r\n" );
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document.write( " y + 96 =  \r\n" );
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document.write( " y - 96 =  \r\n" );
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document.write( "with integer non-negative n and m, and we understand that m < n. \r\n" );
document.write( "Subtracting the lower equation from the upper one, we get\r\n" );
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document.write( " - = 192,\r\n" );
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document.write( "  = 192 = 64*3 =  . (4)\r\n" );
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document.write( "From it, we conclude that m = 6, n-m = 2; hence n-6 = 2, n = 8.\r\n" );
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document.write( "Thus y+96 = =  = 256; hence y = 256-96 = 160.\r\n" );
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document.write( "CHECK: y-96 = =  = 64; hence y = 64+96 =160. <<<---=== Check says OK.\r\n" );
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document.write( "Now from (3) \r\n" );
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document.write( " = (y+96)*(y-96) = (160+96)*(160-96) = 16384 =  ,\r\n" );
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document.write( "Hence, x = 14.\r\n" );
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document.write( "ANSWER. This problem has two solutions (x,y) = (14,160) and (x,y) = (14,-160).\r\n" );
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document.write( "From where the second, negative value of y came ?\r\n" );
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document.write( "Since right side of equation (1) is y^2, it is clear that with positive solution y= 160,\r\n" );
document.write( "negative solution y= -160 works, too.\r\n" );
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document.write( "Where we missed it in our reasoning ? - Because in (4), we could take NEGATIVE factors  and -3 \r\n" );
document.write( "into consideration. It would lead us to the negative value of y.\r\n" );
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document.write( "But since we just caught this second solution, we shouldn't worry anymore.\r\n" );
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document.write( "Thus the problem is solved completely, using strict mathematical reasoning,
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document.write( "and two solutions in integer numbers are found. All necessary explanations are given.\r
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document.write( "Happy learning (!)\r
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