Timeline for Remainders $\quad 1\quad 2\quad $ only

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Jul 1, 2013 at 8:44	comment	added	Waldemar		It could also be of interest to find an upper bound for $n(x)$ which would be an improvement over the following one: $$n(x)\leq \frac{P(x)}{2}+1$$
Jun 27, 2013 at 1:56	comment	added	Włodzimierz Holsztyński		Thank you, @Waldemar, that's pleasing. To celebrate it I'll add the proof of my small theorem to my "Question". On the other hand one still can strive at getting sharper results, e.g. for $x > M$ (say, for $x \ge 19 :-) or other forms.
Jun 26, 2013 at 12:05	comment	added	Waldemar		Thus, referring to your original question, as we know a case for $x=17$ we can conclude that, in general, the inequality $$n(x)\ge\left\lceil\sqrt{P(x)}\right\rceil+1$$ cannot be improved.
Jun 26, 2013 at 11:37	comment	added	Włodzimierz Holsztyński		your first inequality is an interesting conjecture. The second one I had from the beginning, and just somehow neglected to type it in full (just psychology or funny laziness). Yes, it's a theorem, not just a conjecture. Indeed, actual inequality is $$n(x)\ge\left\lceil\sqrt{P(x)}\right\rceil+1$$ Its proof is very simple.
Jun 26, 2013 at 10:42	history	edited	Waldemar	CC BY-SA 3.0	added 45 characters in body
Jun 26, 2013 at 10:22	history	answered	Waldemar	CC BY-SA 3.0