\r\n" );
document.write( "Ikleyn has found two solutions, (x,y) = (5,0) and (x,y) = (0,5) \r\n" );
document.write( "but has failed to show that there are no other solutions. But it is\r\n" );
document.write( "necessary to show that there are no other solutions, not just to\r\n" );
document.write( "find two solutions and verify that they are both solutions.\r\n" );
document.write( "\r\n" );
document.write( "

\r\n" );
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document.write( "Those will be the only solutions if and only if\r\n" );
document.write( "\r\n" );
document.write( "

, for
and
\r\n" );
document.write( "\r\n" );
document.write( "Assume
\r\n" );
document.write( "\r\n" );
document.write( "If x = 0, then 





, except for the solution (x,y) = (0,5)\r\n" );
document.write( "\r\n" );
document.write( "If x = 1, then 







\r\n" );
document.write( "\r\n" );
document.write( "If
then 





\r\n" );
document.write( "\r\n" );
document.write( "The numerator of that last fraction has xy-x indicated factors and the\r\n" );
document.write( "denominator has y factors.\r\n" );
document.write( "\r\n" );
document.write( "Since
and
,\r\n" );
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document.write( "
\r\n" );
document.write( "Subtract 1 from both sides\r\n" );
document.write( "
\r\n" );
document.write( "Divide both sides by x-1\r\n" );
document.write( "
\r\n" );
document.write( "
\r\n" );
document.write( "
\r\n" );
document.write( "
\r\n" );
document.write( "
\r\n" );
document.write( "
\r\n" );
document.write( "
\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "This proves the numerator has at least as many indicated factors as the\r\n" );
document.write( "denominator. The first x factors of the denominator are, respectively, less \r\n" );
document.write( "than or equal to the first x factors of the numerator. The remaining indicated\r\n" );
document.write( "factors of the numerator, if any, are even greater. Therefore, the numerator of\r\n" );
document.write( "the fraction
is greater than the denominator, and the fraction \r\n" );
document.write( "is greater than or equal to 1, and therefore not equal to
.\r\n" );
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document.write( "The case
is proved by symmetry.\r\n" );
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document.write( "Therefore the 2 solutions Ikleyn found above are the ONLY solutions.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "