If you calculate, for m=1,2,3, ...

85^m - (int(85^(m/4)))^4

you get the following:

4
664
82684
0
6287029
1271393704
26172579244
0
32738295586176
246734877110272
9.73675819302912E+015
0

Not surprising, every 4th one is zero, whenever m is 0 mod 4.  So the only hope of the original equation being true is for an 'm' that is 1 mod 4.  And for m=1, this one works perfectly.  But as m gets larger, the differences are much too large to ever be as small as 4.

Not as mathematically rigorous as I'd like, but ...