Apéry's theorem

Apéry's theorem states that the Riemann zeta value ζ(3), also called Apéry's constant, is irrational.

Apéry's proof of the irrationality of ζ(3): Roger Apéry's proof that ζ(3) is irrational, based on rapidly converging rational approximations and recurrence relations. This is the specific argument behind the theorem's conclusion.
Roger Apéry: A French mathematician who proved the irrationality of ζ(3) in 1978. His proof established the theorem bearing his name.
Riemann zeta function: The complex function defined initially by the series sum of n to the power minus s for real part of s greater than one. Apéry's theorem concerns one particular value of this function.
Irrationality of ζ(2): The result that ζ(2) is irrational, following from Euler's evaluation ζ(2)=π²/6. Its elementary route contrasts with the intricate proof required for ζ(3).
Apéry numbers: The integers defined by a finite binomial sum that appear in Apéry's proof of ζ(3)'s irrationality. They supply integer terms in the recurrences used to approximate ζ(3).
1978 International Congress of Mathematicians: The Helsinki meeting of the International Congress of Mathematicians held in 1978. Apéry announced his irrationality result at this congress.
Apéry's constant: The real number ζ(3), equal to the sum of the reciprocals of positive integers cubed. It is the number whose irrationality the theorem proves.
Irrationality of ζ(5): The theorem that the Riemann zeta value ζ(5) is irrational. It is a later odd zeta value whose irrationality is also established.
Apéry recurrence: A second-order recurrence satisfied by Apéry numbers and by a related sequence approximating ζ(3). Its integer coefficients make the rationality contradiction possible.
Henri Cohen: A French mathematician known for work in computational number theory and mathematical software. He contributed to checking and clarifying Apéry's initially surprising proof.

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