The authors prove Rivoal's “denominator conjecture” concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the vector space over \(\mathbb Q\) spanned by \(1,\zeta(m),\zeta(m+2),\dots,\zeta(m+2h)\), where \(m\) and \(h\) are integers such that \(m\ge2\) and \(h\ge0\). In particular, the authors immediately get the following results as corollaries: at least one of the eight numbers \(\zeta(5),\zeta(7),\dots,\zeta(19)\) is irrational, and there exists an odd integer \(j\) between \(5\) and \(165\) such that \(1\), \(\zeta(3)\) and \(\zeta(j)\) are linearly independent over \(\mathbb{Q}\). This strengthens some recent results. The authors also prove a related conjecture, due to Vasilyev, and as well a conjecture, due to Zudilin, on certain rational approximations of \(\zeta(4)\). The proofs are based on a hypergeometric identity between a single sum and a multiple sum due to Andrews. The authors hope that it will be possible to apply their construction to the more general linear forms constructed by Zudilin, with the ultimate goal of strengthening his result that one of the numbers \(\zeta(5),\zeta(7),\zeta(9),\zeta(11)\) is irrational.