# Can $a_1^{k_1}+a_2^{k_2}+\cdots+a_n^{k_n}$ attain every possible residue modulo $m$?

Is it possible to find positive integers $1 < a_1 < a_2 < \cdots < a_n$ such that $$a_1^{k_1}+a_2^{k_2}+\cdots+a_n^{k_n},$$ where the $k_1,k_2,\ldots,k_n$ are nonnegative integers, can attain every possible residue modulo $m$ where $m$ is a positive integer?

For example, if we consider $2^{k_1}+3^{k_2}+5^{k_3}$ then since $$2^{k_1}+3^{k_2}+5^{k_3} \not \equiv 1 \pmod{30},$$ this is not an example of such a set of $a_i$. How can we find such a set of $a_i$ or prove that none exists?

• at least $n >3$. – Ahmad Oct 11 '17 at 3:46
• Do you search the minimal $n$ such that universal numbers $a_i$ exist with this property, if such an $n$ exists ? – Peter Oct 11 '17 at 12:04