Residues

[Zan]

Find all $n\in\mathbb{N}$ so that all residues modulo $n$ can be written in the form $\frac{m(m+1)}{2}$ mod $n$ with $m\in\mathbb{N}_0$.

[Phlembac Adib Hasan]

Nice problem, Zan.Post more and more like these.

[Zan]

So $f(i)\not\equiv0$(mod $n$) if $i\in N_n-0 \ldots$

Could you explain it, please? And did you mean $i\in N_n \backslash0$?

[Phlembac Adib Hasan]

So $f(i)\not\equiv0$(mod $n$) if $i\in N_n-0 \ldots$

Could you explain it, please? And did you mean $i\in N_n \backslash0$?

Yes.Also $N_{2^k}\equiv N_{2^{k+1}}-N_{2^k}(mod\; 2^k)$ means $N_{2^k}\equiv N_{2^{k+1}}\backslash N_{2^k}(mod\; 2^k)$
Like I said before, I was in a hurry and I avoided typing long codes like \backslash.

[Zan]

Thanks!
But could you explain me your argumentation, please?
I'm not very fast

[Phlembac Adib Hasan]

Which part?
At first I showed we need to find those $n$s for which $S_n=\{\frac {0\cdot 1}{2},\frac {1.2}{2},...,\frac {(n-1)n}{2}\}$ form a complete residue system modulo $n$.Hence there must be only one element in the set which is divisible by $n$.As we are seeing $n|\frac {0\cdot 1}{2}$, so no other element in $S_n$ is divisible by $n$.Then I showed $n$ can't be odd because $n|\frac {(n-1)n}{2}$, another element of $S_n$ if $n$ is odd.Next I showed If $n=2^aq$($q$ odd>1), then we can find another element in $S_n$,different from $\frac {0\cdot 1}{2}$, which is divisible by $n$.(For example, if $n=12$ then $12|\frac {8\times 9}{2}$.)At last I proved by induction that each positive integer power of $2$ satisfies the conditions.

[Zan]

The part (i) with a lot of dots.

[Phlembac Adib Hasan]

I hate formal solution writing because it makes everything crazy. Notice that for each $i\in N_n$,\[i\equiv f(b_{n_{i}})(mod\; n).\]So $\{f\left (b_{n_{0}}\right ),f\left (b_{n_{1}}\right ),...,f\left (b_{n_{n-1}}\right )\}$ makes complete residue set modulo $n$.Notice that all $b_{n_{i}}$s must be distinct modulo $n$ to follow the last condition.So for each $b_{n_{i}}$ there exists a unique $a_{n_{b_{n_{i}}}}\in N_n$ from the first definition.Hence it follows that \[f\left (b_{n_{i}}\right )\equiv f\left (a_{n_{b_{n_{i}}}}\right )(mod\; n)\].So it follows that $S_n$ is reduced residue system modulo $n$.Last part of $(i)$ is explained in my previous post.Let me know whehter it is clear enough now.

[Zan]

Thank you very much!

[sourav das]

I want to give another solution:

Actually given condition implies that we need to find those $n$ such that set $A=\{x|x=\frac{m(m+1)}{2}$ and $m\in \mathbb N_0$ and $0\leq m \leq n-1\}$ is a complete residue set modulo $n$.

We claim all $n=2^k$ works where $k\in \mathbb N_0$
First let us prove $n>1$ odd won't work
Proof:
It's simple as for $m=0,n-1$ , both are congruent to $0$ mod $n$ as $n$ odd

It implies $n$ cannot have any odd prime factor.
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Now , for the contrary of our claim let there exist $n,m$ non- congruent mod $2^k$ such that $\frac{n(n+1)}{2}=\frac{m(m+1)}{2}$ $(mod 2^k) $,Implies $\frac{(m-n)(m+n+1)}{2}=0$ ($mod$ $2^k$) .as only one of $\frac{m-n}{2},\frac{m+n+1}{2}$ is even. It implies $2^k|\frac{m+n+1}{2}<2^k$ So, contradiction. (Proved)