Decimations of $\ell$-Sequences and Permutations of Even Residues $\modp$
Jean Bourgain, Todd Cochrane, Jennifer Paulhus, Christopher G. Pinner · SIAM Journal on Discrete Mathematics · 2009
Goresky and Klapper conjectured that for any prime $p>13$ and any $\ell$-sequence ${\bf a}$ based on p, every pair of allowable decimations of ${\bf a}$ is cyclically distinct. The conjecture is essentially equivalent to the statement that the mapping $x\to Ax^d$, with $(d,p-1)=1$, $p mid A$, is a permutation of the even residues $\!\pmod p$ if and only if $d=1$ and $A\equiv1\pmod p$ for $p>13$. We prove the conjecture for $p>2.26\cdot10^{55}$ and establish it in a number of other special cases such as when $-.000274p