document.write( "Question 1209939: If m is a positive integer and sqrt(4m^2+29) is an integer, then what is m? \n" ); document.write( "

Algebra.Com's Answer #851126 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "If m is a positive integer and sqrt(4m^2+29) is an integer, then what is m?
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document.write( "If     is an integer number n, then\r\n" );
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document.write( "    4m^2 + 29 = n^2\r\n" );
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document.write( "    29 = n^2 - 4m^2\r\n" );
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document.write( "    29 = (n-2m)*(n+2m).\r\n" );
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document.write( "29 is a prime number, so it has two possible decompositions into the product of prime factors \r\n" );
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document.write( "        29 = 1* 29   or  29 = (-1)*(-29).\r\n" );
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document.write( "Therefore, we have 4 systems of linear equations to analyze\r\n" );
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document.write( "    n - 2m =  1,    (1)\r\n" );
document.write( "    n + 2m = 29,    (2)\r\n" );
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document.write( "    n - 2m =  -1,   (3)\r\n" );
document.write( "    n + 2m = -29,   (4)\r\n" );
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document.write( "    n - 2m =  29,   (5)\r\n" );
document.write( "    n + 2m =   1,   (6)\r\n" );
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document.write( "    n - 2m = -29,   (7)\r\n" );
document.write( "    n + 2m =  -1.   (8)\r\n" );
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document.write( "From system (1), (2), by subtracting equations, we have \r\n" );
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document.write( "    2m - (-2m) = 29-1,  4m = 28,  m = 28/4 = 7  and then  n = 1 + 2m = 1 + 2*7 = 15.\r\n" );
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document.write( "So, the solution pair is (m,n) = (7,15), and it works properly: \r\n" );
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document.write( "     =  =  = 15.\r\n" );
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document.write( "From system (3), (4), by subtracting equations, we have \r\n" );
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document.write( "    2m - (-2m) = -29-(-1),  4m = -28,  m = -28/4 = -7.  \r\n" );
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document.write( "It does not work, since the number m should be positive, by the condition.\r\n" );
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document.write( "From system (5), (6), by subtracting equations, we have \r\n" );
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document.write( "    2m - (-2m) = 1-29,  4m = -28,  m = -28/4 = -7.  \r\n" );
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document.write( "It does not work, since the number m should be positive, by the condition.\r\n" );
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document.write( "From system (7), (8), by subtracting equations, we have \r\n" );
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document.write( "    2m - (-2m) = -1 - (-29),  4m = 28,  m = 28/4 = 7  and then  n = 1 + 2m = 1 + 2*7 = 15.\r\n" );
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document.write( "So, the solution pair is (m,n) = (7,15), the same as we got from system (1), (2), \r\n" );
document.write( "and it works properly:   =  =  = 15.\r\n" );
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document.write( "ANSWER.  For the given problem, there is a unique solution for m.  It is m = 7.\r\n" );
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\n" ); document.write( "\n" ); document.write( "I placed this my solution here after the solution by tutor @math_tutor2020 to make the analysis complete.\r
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\n" ); document.write( "\n" ); document.write( "My comment/response to the Edwin's comment, regarding my post.\r
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document.write( "    For this given problem, considering all four decompositions\r\n" );
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document.write( "        29 = 1*29 = 29*1 = (-1)*(-29) = (-29)*(-1)\r\n" );
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document.write( "    is NECESSARY for the completeness of the analysis.\r\n" );
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document.write( "    So, all 4 (four) cases/decompositions MUST be considered, exactly as it was made in my post.\r\n" );
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