A similar argument holds for two numbers which are 3 times a perfect square. Therefore the assumption that m is a perfect square does not hold. We have two cases. Assume our number is of the form A4C258G6I0. Assume our number is of the form A8C654G2I0.
T is the carry out of that position.
D' to refer to a digit D on the left side of a string with a corresponding digit D' on the right side of the string. This enables us to construct a finite state machine to recognize those palintiples. I have written a computer program that constructs finite state machines for recognizing palintiples for various bases and constants. I found that base 10 is actually an unusually boring base for this problem. If anyone investigates the general behavior, I would be most happy to hear about it.