fixed c, δ > 0. The starting point for Apéry was the series representation of ζ(3) as Roughly speaking, Apéry then defined a sequence cn,k which converges to ζ(3) about as fast as the above series, specifically He then defined two more sequences an and bn that, roughly, have the quotient cn,k. These sequences were and The sequence an/bn converges to ζ(3) fast enough to apply the criterion, but unfortunately an is not an integer after n = 2. Nevertheless, Apéry showed that even after multiplying an and bn by a suitable integer to cure this problem the convergence was still fast enough to guarantee irrationality. Later proofs Within a year of Apéry's result an alternative proof was found by Frits Beukers, who replaced Apéry's series with integrals involving the shifted Legendre polynomials . Using a representation that would later be generalized to Hadjicostas's formula, Beukers showed that for some integers An and Bn (sequences and ). Using partial integration and the assumption that ζ(3) was rational and equal to a/b, Beukers eventually derived the inequality which is a contradiction since the right-most expression tends to zero as n → ∞, and so must eventually fall below 1/b. A more recent proof by Wadim Zudilin is more reminiscent of Apéry's original proof, and also has similarities to a fourth proof by Yuri Nesterenko. These later proofs again derive a contradiction from the assumption that ζ(3) is rational by constructing sequences that tend to zero but are bounded below by some positive constant. They are somewhat less transparent than the earlier proofs, since they rely upon hypergeometric series. Higher zeta constants See also Apéry and Beukers could simplify their proofs to work on ζ(2) as well thanks to the series representation Due to the success of Apéry's method a search was undertaken for a number ξ5 with the property that If such a ξ5 were found then the methods used to prove Apéry's theorem would be expected to work on a proof that ζ(5) is irrational. Unfortunately, extensive computer searching has failed to find such a constant, and in fact it is now known that if ξ5 exists and if it is an algebraic number of degree at most 25, then the coefficients in its minimal polynomial must be enormous, at least 10383, so extending Apéry's proof to work on the higher odd zeta constants does not seem likely to work.