On a sequence of polynomials
For any integer \(k\geq 2\) let \(p_k\) be a polynomial defined by \(p_k(X)= (X+1)^k -X^k-1\). It is easy to see that \(p_k\) is divisible by \(X\), \(X(X+1)\), \(X(X^2 +X+1)\) and \(X(X+1)\) \((X^2+ X+1)^2\) if \(k\) is even, \(k\equiv 3\pmod 6\), \(k\equiv 5\pmod 6\) and \(k\equiv 1 \pmod 6\), respectively. In connection with studies of solutions to the Korteweg-de Vries equation, J. Sanders asked if, for any \(l>k>1\), \(\text{gcd} (p_k,p_l)\) contains only the trivial divisors mentioned above. In this paper the author proposes a positive reply to this question.
scientific article; zbMATH DE number 1025691