Determine all possible integer solutions, whether positive or negative, to this equation:
5y^{4} + 2560y^{2} = x^{5}  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 20060315 23:59:42 