document.write( "Question 1106492: m^2+n^2=-mn
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Algebra.Com's Answer #721482 by ikleyn(53765) $\"\"$ $\"About$

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\n" ); document.write( "No.\r
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\n" ); document.write( "\n" ); document.write( "Counter-example: m = 1, n = 1.\r
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\n" ); document.write( "By the way, what we see in the post, is NOT expression.\r
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\n" ); document.write( "\n" ); document.write( "It is an equation, or an equality, or an (hypothetical) identity . . . \r
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\n" ); document.write( "\n" ); document.write( "I suspect that the formulation in the post is NOT that you wanted to know/to ask.\r
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\n" ); document.write( "\n" ); document.write( "More meaningful questions are:\r
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document.write( "    1)  Can this equality be true for all pairs (m,n) of integer numbers?\r\n" );
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document.write( "    2)  Can this equality be true for some pairs (m,n) of integer numbers? At least for one pair ? \r\n" );
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document.write( "        (providing m=/= 0, n=/= 0 in these pairs (in this pair) ).\r\n" );
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document.write( "Solution\r\n" );
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document.write( "1)  Can this equality be true for all pairs (m,n) of integer numbers?\r\n" );
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document.write( "    I just answered this question above by giving a counter-example: No, it can not.\r\n" );
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document.write( "2)  Can this equality be true for some pairs (m,n) of integer numbers?\r\n" );
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document.write( "    m^2 + n^2 = -mn.\r\n" );
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document.write( "    Again, the answer is NO.\r\n" );
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document.write( "    a)  if m and n  are both odd numbers, then the left side is an even integer number, while the right side is an odd integer.\r\n" );
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document.write( "        Contradiction.\r\n" );
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document.write( "    b)  if m is an even integer and n is an odd, then the left side is an odd, while the right side is an even.\r\n" );
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document.write( "        Contradiction.\r\n" );
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document.write( "    c)  if m is an odd integer and n is an even, then we have the same contradiction.\r\n" );
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document.write( "    d)  If both m and n are even, then in the given equality we can reduce both m and n by dividing each by 2, and we will repeat it \r\n" );
document.write( "        until we get one of the three cases a), b) or c).\r\n" );
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document.write( "        Then exactly as in each of these cases we will get the same contradiction.\r\n" );
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