{"schema":"vela.problem-packet.v0.1","problem":820,"statement":"Let $H(n)$ be the smallest integer $l$ such that there exist $k<l$ with $(k^n-1,l^n-1)=1$.Is it true that $H(n)=3$ infinitely often? (That is, $(2^n-1,3^n-1)=1$ infinitely often?)Estimate $H(n)$. Is it true that there exists some constant $c>0$ such that, for all $\\epsilon&#62;0$,\\[H(n) &#62; \\exp(n^{(c-\\epsilon)/\\log\\log n})\\]for infinitely many $n$ and\\[H(n) &#60; \\exp(n^{(c+\\epsilon)/\\log\\log n})\\]for all large enough $n$?Does a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$?","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A263647","name":"Numbers k such that 2^k-1 and 3^k-1 are coprime.","terms":"1,2,3,5,7,9,13,14,15,17,19,21,25,26,27,29,31,34,37,38,39,41,45,47,49,51,53,57,59,61,62,63,65,67,71,73,74,79,81,85,87,89,","url":"https://oeis.org/A263647"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}