I'm looking at OEIS:A000236, whose definition states:
Maximum m such that there are no two adjacent elements belonging to the same n-th power residue class modulo some prime p in the sequence 1,2,...,m (equivalently, there is no n-th power residue modulo p in the sequence 1/2,2/3,...,(m-1)/m).
I can't quite understand this. Here's what I have so far:
- The power residue class $b=[a]_m$ represents all $b$ for which $a=b \pmod m$ (thanks to Inceptio's answer). This represents all possible remainders of $a \div m$.
- A necessary and sufficient condition for $t \in R$ where $R$ consists of the $n$th power residues $\pmod p$ is that $x^k \equiv t \mod p$ is solvable ($x$ exists). Source: CMS
- If $z$ is the maximum value such that there are no two adjacent elements in 1, 2, ..., m belonging to the same power residue class, then that means the $z$ and $z+1$ belong to the same power residue class
This leads me to believe that if two numbers $z_1, z_2$ are in the same $n$th power residue class $\pmod p$, it means that there exist $y_1, y_2$ such that $y_1^n \equiv z_1 \pmod p$ and $y_2^n \equiv z_2 \pmod p$
If this is correct, great. If it is incorrect, could someone explain to me where my ideas are wrong and what this sequence means? Thanks!