Apéry's Theorem

Theorem

Apéry's constant:

$\map \zeta 3 = \ds \sum_{n \mathop = 1}^\infty \frac 1 {n^3}$

is irrational.

Proof

We have:

$\dfrac 6 {\map \zeta 3} = 5 - \cfrac {1^6} {117 - \cfrac {2^6} {535 - \cfrac {\dotsb} {\dotsb - \cfrac {n^6} {34 n^3 + 51 n^2 + 27 n + 5} - \dotsb } } }$

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Also known as

Some sources present the name of this without the accent: Apery's theorem.

Source of Name

This entry was named for Roger Apéry.

Sources

• 1979: Roger Apéry: Irrationalité de $\map \zeta 2$ et $\map \zeta 3$ (Astérisque Vol. 61: pp. 11 – 13)
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1,20205 69 \ldots$
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1 \cdotp 202 \, 056 \ldots$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Apery's theorem
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1 \cdotp 20205 \, 6 \ldots$

• Weisstein, Eric W. "Apéry's Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AperysConstant.html