(In reply to
computer exploration by Charlie)
There doesn't seem to be much to go on here. The solutions look like coincidences rather than a pattern.
The last two coming from
(3^31)=2*13
(313^31)=2^3*3^2*13*181^2
filling just the right gaps so the product is
(2^2*3*13*181)^2
and
(20^3+1)=3^2*7*127
(362^3+1)=3^2*7^3*11^2*127
product
(3^2*7^2*11*127)^2
Other choices for x leave factors that may or may not ever be filled by some y. For example
(11^31)=2*5*7*19
Does some large y give a number have these prime factors to odd powers and any others to even powers?
Using y=2*5*7*19+1 gives each factor (difference of cubes) but doesn't guarantee powers.

Posted by Jer
on 20180228 15:19:12 