Skip to main content

On Catalan Constant Continued Fractions

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

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

  • 307 Accesses


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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others


  1. 1.

  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.


  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to David Naccache .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

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.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-33016-2

  • Online ISBN: 978-3-031-33017-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics