document.write( "Question 1162885: For how many integers n is n^2 + 18n + 13 a perfect square? \n" ); document.write( "

Algebra.Com's Answer #786805 by ikleyn(52832) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " It looks AMAZINGLY, but there is the way to solve the problem formally, in strict Algebra logic, without trials and errors.\r
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document.write( "Let  n^2 + 18n + 13 = m^2   be a perfect square.\r\n" );
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document.write( "Then\r\n" );
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document.write( "    (n + 9)^2 - 68 = m^2\r\n" );
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document.write( "    (n + 9)^2 - m^2 = 68\r\n" );
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document.write( "    (n + m + 9)*(n - m + 9) = 68.\r\n" );
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document.write( "Decompositions for 68 are  1*68, 2*34, 4*17, 17*4, 34*2 and 68*1.\r\n" );
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document.write( "For each decomposition, we have the system of equations\r\n" );
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document.write( "    n + m + 9 =  1\r\n" );
document.write( "    n - m + 9 = 68\r\n" );
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document.write( "    n + m + 9 =  2\r\n" );
document.write( "    n - m + 9 = 34\r\n" );
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document.write( "    n + m + 9 =  4\r\n" );
document.write( "    n - m + 9 = 17\r\n" );
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document.write( "    n + m + 9 = 17\r\n" );
document.write( "    n - m + 9 =  4\r\n" );
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document.write( "    n + m + 9 = 34\r\n" );
document.write( "    n - m + 9 =  2\r\n" );
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document.write( "    n + m + 9 = 68\r\n" );
document.write( "    n - m + 9 =  1\r\n" );
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document.write( "Easy analysis shows that some of these systems produce non-integer solution.\r\n" );
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document.write( "The only system, which produces appropriate integer solution, is THIS\r\n" );
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document.write( "    n + m + 9 = 34\r\n" );
document.write( "    n - m + 9 =  2\r\n" );
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document.write( "The solution is  n = 9, m = 16.\r\n" );
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document.write( "Therefore,  n = 9 is the solution to the problem.\r\n" );
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document.write( "If you analyze the similar systems with decomposition of the number 68 into the product of negative factors, \r\n" );
document.write( "you will find another solution n = -27.\r\n" );
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document.write( "In all, there are 2 (two) such numbers n, 9 and -27.\r\n" );
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