Number theory involving sum of digits of a number:

Question

Let a fixed natural number m be given Call a positive integer n to be a GRT number iff:

1. $n \equiv 1 \pmod m$

2. Sum of digits in decimal representation of $n^2$ is greater than or equal to sum of digits in decimal representation of $n$.

How many GRT numbers are there ?

Answer 1

I can't prove it absolutely, but for any $m$ there must be infinitely many. Any large $n$ that is $\equiv 1 \bmod m$ will have a square with lots more digits than $n$ has, so the sum of digits of the square will be larger.