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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 ?

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    $\begingroup$ 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. $\endgroup$ – dan_fulea Feb 1 at 17:34
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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.

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