Can You Solve This Complex Integer Equation System?

[elimqiu]

Find smallest $$d=2a^2=3b^3+2=5c^5+3$$
where a,b,c are integers.
This problem is very hard. Either no solution or d is toooooooo big

 

[The Chaz]

well, Chuck...?

 

[CRGreathouse]

I've actually done a good bit of work on this, see here:
http://mymathforum.com/viewtopic.php?f=40&t=13843