I have a task:
Find all $n\ \epsilon \ N, \ n > 1$ for which a permutation $a_1,\ a_2,\ ...,\ an\ $ of numbers $ 0,1, ..., n - 1$ exists such that $a_1,\ a_1+a_2,\ ...,\ a_1+a_2+\ ...\ +an\ $ form a CRS $mod\ n$.
So far I've come to the conclusion that $a_1$ must be $0$ because otherwise there would be two equal numbers from the listed above (so it won't be a CRS) and I think that odd numbers don't form CRS because $n*(n-1)/2 \ \ (mod\ n)$ is also $0$ so again won't form CRS. My assumption is that all even numbers can form CRS and satisfy the conditions above since I have found such permutations for numbers 2,4,6,8, but I don't know how to prove it and if it is right at all.
Would appreciate some help: first - whether I have made any mistakes so far, and second - with proof if I am right.
Thanks in advance!