Article
An Elementary Beukers-Like Proof for the Irrationality of $π^2$
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Abstract
In a pioneering work [Bull. London Math. Soc. 11, 268 (1979)], Beukers introduced a simple method, based upon double integrals over the unit-square $[0, 1] \times [0, 1]$ involving Legendre polynomials, for proving the irrationality of $\zeta(2)$ and $\zeta(3)$. In a more recent paper [Amer. Math. Monthly 108, 222 (2001)], Huylebrouck has adapted the Beukers method for proving the irrationality of $\pi$, among other mathematical constants. In this note, we modify Huylebrouck's proof in order to show the stronger result that $\pi^2$ is irrational using only single integrals over $[0, 1]$.
Keywords
Irrationality proof, Beukers method, Legendre polynomials
Citation
Lima, F. M. S. & Jordão, P. G. F. (2019). An elementary beukers-like proof for the irrationality of $π^2$. Journal of Analysis & Number Theory, 7(2), 35–37. https://doi.org/10.18576/jant/070202
F. M. S. Lima and P. G. F. Jordão, “An elementary beukers-like proof for the irrationality of $π^2$,” Journal of Analysis & Number Theory, vol. 7, no. 2, pp. 35–37, 2019, doi: 10.18576/jant/070202.
Lima FMS, Jordão PGF. An elementary beukers-like proof for the irrationality of $π^2$. Journal of Analysis & Number Theory. 2019;7(2):35–37. doi:10.18576/jant/070202.
Lima, F. M. S. and Jordão, P. G. F. (2019), ‘An elementary beukers-like proof for the irrationality of $π^2$’, Journal of Analysis & Number Theory, 7(2), pp. 35–37. Available at: https://doi.org/10.18576/jant/070202.
Lima, Fábio M. S., and Pedro G. F. Jordão. “An Elementary Beukers-like Proof for the Irrationality of $π^2$.” Journal of Analysis & Number Theory, vol. 7, no. 2, 2019, pp. 35–37. https://doi.org/10.18576/jant/070202.
Lima, Fábio M. S., and Pedro G. F. Jordão. “An Elementary Beukers-like Proof for the Irrationality of $π^2$.” Journal of Analysis & Number Theory 7, no. 2 (2019): 35–37. https://doi.org/10.18576/jant/070202.
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