document.write( "Question 67831: Find all pairs (m, n) of positive integers such that m2/(2mn2 - n3 + 1) is a positive integer. \r
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Algebra.Com's Answer #48279 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
Find all pairs (m, n) of positive integers such \r\n" );
document.write( "that m²/(2mn² - n³ + 1) is a positive integer. \r\n" );
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document.write( "background of the problem ..some preliminary concepts and review of\r\n" );
document.write( "related literature\r\n" );
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document.write( "       m²\r\n" );
document.write( "———————————————\r\n" );
document.write( " 2mn² - n³ + 1\r\n" );
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document.write( "Case 1.  When the denominator = 1, then the above fraction is m², which\r\n" );
document.write( "is a positive integer\r\n" );
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document.write( "The denominator = 1 is true when and only when\r\n" );
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document.write( " 2mn² - n³ + 1 = 1, which is true when and only when\r\n" );
document.write( "     2mn² - n³ = 0, which is true when and only when\r\n" );
document.write( "    n²(2m - n) = 0\r\n" );
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document.write( "n is not 0 since it is a positive integer, so the last\r\n" );
document.write( "equation is true when and only when\r\n" );
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document.write( "        2m - n = 0, which is true when and only when\r\n" );
document.write( "            2m = n\r\n" );
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document.write( "So case 1 tells us that the given expression is a positive\r\n" );
document.write( "integer when n = 2m.  So the ordered pair (2m,m) will always\r\n" );
document.write( "result in a positive integer for the given expression.\r\n" );
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document.write( "Case 2.  When n = 1, the last two terms cancel\r\n" );
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document.write( "       m²\r\n" );
document.write( "——————————————— =\r\n" );
document.write( " 2mn² - n³ + 1         \r\n" );
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document.write( "         m²\r\n" );
document.write( "——————————————————— =\r\n" );
document.write( " 2m(1)² - (1)³ + 1    \r\n" );
document.write( "       \r\n" );
document.write( "        m²\r\n" );
document.write( "       ———— =\r\n" );
document.write( "        2m\r\n" );
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document.write( "         m\r\n" );
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document.write( "         2\r\n" );
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document.write( "And if m is chosen even, say m = 2k,\r\n" );
document.write( "then the expression will be an integer,\r\n" );
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document.write( "So pairs (m,n) = (2k,1)\r\n" );
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document.write( "When n = 2\r\n" );
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document.write( "       m²\r\n" );
document.write( "——————————————— =\r\n" );
document.write( " 2mn² - n³ + 1 \r\n" );
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document.write( "         m²\r\n" );
document.write( "——————————————————— =\r\n" );
document.write( " 2m(2)² - (2)³ + 1 \r\n" );
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document.write( "      m²\r\n" );
document.write( "———————————— =\r\n" );
document.write( " 8m - 8 + 1 \r\n" );
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document.write( "    m²\r\n" );
document.write( "———————— =\r\n" );
document.write( " 8m - 7 \r\n" );
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document.write( "Divide numerator and denominator\r\n" );
document.write( "by m\r\n" );
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document.write( "     m\r\n" );
document.write( "————————— =\r\n" );
document.write( " 8 - 7/m \r\n" );
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document.write( "This is an integer when m = 1 or 7, so\r\n" );
document.write( "we have solutions (2,7) and (2,1).  (2,1)\r\n" );
document.write( "is a solution form case 1, so the only new\r\n" );
document.write( "solution we have is (n,m) = (2,7)\r\n" );
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document.write( "My computer found another solution\r\n" );
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document.write( "(n,m) = (4, 126) \r\n" );
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document.write( "and I'm not sure how to get it.  Sorry!\r\n" );
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document.write( "What math course is this?  Number theory?\r\n" );
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document.write( "But apparently all the solutions are:\r\n" );
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document.write( "(2m,m), (2k,1), (2,7), and (4, 126)\r\n" );
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document.write( "where m, and k can be any positive integers.\r\n" );
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document.write( "Edwin
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