Find a numeric coincidence when representing numbers in various bases
Python 2, 158 bytes

Came a bit too easily, so I might be missing something obvious!

s=lambda n,b:(n/b and s(n/b,b)or'')+chr(48+n%b+39*(n%b&gt;9))
def f(a,b,c):
 for n in range(b,36**4):
  x=s(n,a);y=s(n,b)
  if int(x,b)==int(y,c):print x,y;break


The base conversion function is lovingly stolen from @ugoren's solution to Print integers in any base up to 36. The code simply tries all possible values for the smaller number and the rest is fully determined. We start counting from base2 to ensure both sequences are at least length 2.

&gt;&gt;&gt; f(8, 10, 16)
100 64
&gt;&gt;&gt; f(8, 10, 17)
&gt;&gt;&gt; f(16, 20, 24)
20 1c