# Number theory involving sum of digits of a number:

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 ?

• Please fix the question and please never type something like $2. Sum \ of \ digits \ in \ decimal \ representation \ of \ n^2 \ is \ greater \ than \ or \ equal \ to \ sum \ of \ digits \ in \ decimal \ representation \ of \ n.$ again. – dan_fulea Feb 1 at 17:34

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.