Vanishing sums of powers modulo n - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:36:02Z http://mathoverflow.net/feeds/question/107895 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107895/vanishing-sums-of-powers-modulo-n Vanishing sums of powers modulo n Anton Klyachko 2012-09-23T10:12:52Z 2012-09-23T10:18:40Z <p>Is it possible to describe all positive integer sequences ${x_i}$ such that $$\sum_i x_i=n \quad \hbox{and} \quad \sum_i x_i^k\equiv 0\pmod n \quad \hbox{for all k (and a given n)?}$$</p> <h2>Background</h2> <p>We say that two elements of a group belong to the same <em>tribe</em> if their squares are equal. Then </p> <p><i> the sum of $k$</i>th <i>powers of tribe sizes is divisible by the order of the group for any positive integer $k$</i> [<a href="http://arxiv.org/abs/1205.2824" rel="nofollow">http://arxiv.org/abs/1205.2824</a>, Example 1].</p> <p>This remains true if we replace squares (in the definition of tribes) with cubes or any other powers.</p>