Diophantine equations and identities (Q1072583)

Let \(d(m;x_ 1,...,x_ n)\) be a determinant of the nth order obtained from the circulant \(C(x_ 1,...,x_ n)\) by multiplying all entries below the main diagonal by m. Infinitely many integer solutions \(x_ 1,...,x_ n\) of the equation \((1)\quad d(m;x_ 1,...,x_ n)=\pm 1\) are stated in this paper as terms of certain linear recursive sequences for \(n=2,3,5\) and \(m=D^ n+1\) with \(D\neq 0\) an integer. From this some combinatorial identities are derived. Reviewer's remark: \textit{L. Bernstein} [Trans. Am. Math. Soc. 127, 76-89 (1967; Zbl 0167.316)] gave an infinite system of solutions of (1) without any restrictions on n which coincides with the above result for \(n=2,3\) and 5.