This has been (renamed and) refocused on boboquack’s unforeseen solution, which deserves its own puzzle.
Right when you thought the good old days of
along comes an unending lesson on similarities of
numbers in different bases.
1008 means the digits 100 in
which equals the familiar 64
(in base 10),
which itself is represented as 6410.
Just when is 100 the same as 64? When isn’t it?!   Just look:
1008  = 6410
10010  = 6416 ?   ✓
10016  = 6442 ?   ✓   Could this go on forever?
That used 6 different digits in all — 0, 1, 2, 4, 6, 8 — and may be summarißed as...
K  W  = L X
K X   = L Y
K Y = L Z
...where   K = 100,   L = 64,   W = 8,   X = 10,   Y = 16   and   Z = 42.
And no, this pattern does not continue forever, not even past L Z ( 6442 ).
But wait, other patterns can indeed go on forever.   And with fewer different digits.
M P   = N Q
M Q = N R
M R = N S
M S   = N T
What pattern of M, N, P, Q, R, S, T, . . .   uses the fewest different digits?
How can that be generalized to other numbers of different digits?
( M > N   and   P < Q < R < S < T <   · · · )
To quote professor Tom Lehrer from New Math YouTube, “Hooray for New Math!”