Erdős Problem #820

\(H(n)\) is the smallest integer \(l\) such that there is some \(k < l\) with \((k^n-1, l^n-1) = 1\). We take \(k\) and \(l\) to be integers with \(k \geq 2\), so that \(k^n - 1 \geq 1\) for \(n \geq 1\). Smaller \(k\) give degenerate values: over the integers \(k = 0\) gives \((-1, l^n-1) = 1\), and \(k = 1\) gives \((0, l^n-1) = l^n-1\), which is \(1\) for \(n = 1\), \(l = 2\). Both would contradict the value \(H(1) = 3\) listed in the source. For \(n \geq 1\) the set is nonempty (see `Erdos820.H_spec`). For \(n = 0\) it is empty, a

\(K(n)\) is the smallest integer \(k \geq 2\) such that \((k^n-1, 2^n-1) = 1\). As for `Erdos820.H`, smaller \(k\) are excluded because they give degenerate values. For \(n \geq 2\) the value \(k = 2\) never works, and \(k = 2^n - 1\) always works. For \(n = 1\) we get \(K(1) = 2\), and for \(n = 0\) the set is empty, so `K 0 = 0`.