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Flint Hills Series

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FlintHillSeries1

The Flint Hills series is the series

S_1=sum_(n=1)^infty(csc^2n)/(n^3)

(Pickover 2002, p. 59). Convergence has not been established, since csc^2n can have sporadic large values. The plots above show its behavior up to n=10^4. The positive integer values of n giving incrementally largest values of |cscn| are given by 1, 3, 22, 333, 355, 103993, ... (OEIS A046947), which are precisely the numerators of the convergents of pi, corresponding to the values 1.1884, 7.08617, 112.978, 113.364, 33173.7, ....

Alekseyev (2011) has shown that the question of the convergence of the Flint Hills series is related to the irrationality measure of pi, and in particular, convergence would imply mu(pi)<=2.5, which is much stronger than the best established upper bound.

Meiburg (2022) proved the near-converse that mu(pi)<2.5 implies convergence. The boundary case mu(pi)=2.5 is not settled by these implications.

A purported proof of convergence was given by Mantzakouras and López Zapata (2025). López Zapata (2026) withdrew the manuscript on Sep. 14, 2026 after identifying serious errors in the argument.

OpenAI (2026) announced mu(pi)=2 and, as a consequence, convergence of the Flint Hills series. The release attributes the proof to an unreleased internal OpenAI model and supplies a Lean formalization of the claimed irrationality measure. As of Oct. 7, 2026, the formal statement had been read by VibeMathed (2026), but an independent rebuild and specialist verification of the complete proof had not been reported.

See also

Cookson Hills Series, Irrationality Measure, Tanc Function

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References

Alekseyev, M. A. "On Convergence of the Flint Hills Series." 27 Apr 2011. https://arxiv.org/abs/1104.5100.López Zapata, C. "Diophantine Flint-Hills Series." Withdrawal notice. 14 Sep 2026. https://arxiv.org/abs/2502.03474v2.Mantzakouras, N. and López Zapata, C. "Diophantine Flint-Hills Series: Convergence, Polylogarithms and pi's Bound." 21 Jan 2025. Withdrawn 14 Sep 2026. https://arxiv.org/abs/2502.03474v1.Meiburg, A. "Bounds on Irrationality Measures and the Flint-Hills Series." 29 Aug 2022. https://arxiv.org/abs/2208.13356.OpenAI. "The Irrationality Exponent of pi Is 2." Sep. 24, 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf.Pickover, C. A. "Flint Hills Series." Ch. 25 in The Mathematics of Oz: Mental Gymnastics from beyond the Edge. New York: Cambridge University Press, pp. 57-59 and 265-268, 2002.Sloane, N. J. A. Sequence A046947 in "The On-Line Encyclopedia of Integer Sequences."VibeMathed. "The Irrationality Exponent of pi Is 2." Oct. 7, 2026. https://vibemathed.com/problem/irrationality-exponent-of-pi.

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Flint Hills Series

Cite this as:

Weisstein, Eric W. "Flint Hills Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FlintHillsSeries.html

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