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On Catalan Constant Continued Fractions

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  • First Online:
Codes, Cryptology and Information Security (C2SI 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13874))

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Abstract

The Ramanujan Machine project detects new expressions related to constants of interest, such as \(\zeta \) function values, \(\gamma \) and algebraic numbers (to name a few).

In particular the project lists a number of conjectures concerning the Catalan constant \(G= 0.91596559\ldots \)

We show how to generate infinitely many.

We used an ad hoc software toolchain and rather tedious mathematical developments.

Because we do not provide a proper peer-reviewed proof of the relations given here we do not claim them to be theorems.

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Notes

  1. 1.

    http://www.ramanujanmachine.com/wp-content/uploads/2020/06/catalan.pdf.

  2. 2.

    A typical example is \(\kappa =0\) and \(c=5\) where our formulae predict \(3628800/(1285371-1786050G)\), which, after division by 9, yields the reduced value \(403200/(142819-198450 G)\).

  3. 3.

    With \(\kappa =0\) being an exception that does not fit this general rule.

  4. 4.

    Modulo a possible sign flip for the very first values.

  5. 5.

    By any analytic or computational manner.

  6. 6.

    There is no intrinsic difficulty to keep computing those constants, we limit ourselves to 7 per category for space reasons.

References

  1. Cação, I., Falcão, M., Malonek, H.: Hypercomplex polynomials, Vietoris’ rational numbers and a related integer numbers sequence. Complex Anal. Oper. Theory 11, 06 (2017)

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  2. David, N.B., Nimri, G., Mendlovic, U., Manor, Y., Kaminer, I.: On the Connection Between Irrationality Measures and Polynomial Continued Fractions (2021)

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  3. Raayoni, G., et al.: Generating conjectures on fundamental constants with the Ramanujan machine. Nature 590(7844), 67–73 (2021)

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  4. Raayoni, G., et al.: The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants. CoRR, abs/1907.00205 (2019)

    Google Scholar 

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Correspondence to David Naccache .

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Naccache, D., Yifrach-Stav, O. (2023). On Catalan Constant Continued Fractions. In: El Hajji, S., Mesnager, S., Souidi, E.M. (eds) Codes, Cryptology and Information Security. C2SI 2023. Lecture Notes in Computer Science, vol 13874. Springer, Cham. https://doi.org/10.1007/978-3-031-33017-9_3

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  • DOI: https://doi.org/10.1007/978-3-031-33017-9_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-33016-2

  • Online ISBN: 978-3-031-33017-9

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