Determine all possible integer solutions, whether positive or negative, to this equation:

5y⁴ + 2560y² = x⁵ - 65536

I used a spreadsheet, put in integer values for y, and then plotted the fractional part of the resulting x value.   Graphing this out (y vs fractional part of x)  shows a pattern of arcs and curves that generally all cross the zero line (if = zero then x would be an integer) with a nonzero slope. 

All except one curve which was shaped sort of like the graph of y=x^3 where it crosses the origin.  Anyway, that one curve crossed the zero line such that the local slope was flat or nearly flat, and that was when y=16.

This makes me suspect that 16 is the only answer for x.  (But certainly doesn't prove it)

Posted by Larry on 2006-03-15 23:59:42