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International Conference on Interactive Theorem Proving

ITP 2016: Interactive Theorem Proving pp 140–150Cite as

Proof Pearl: Bounding Least Common Multiples with Triangles

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9807))

Abstract

We present a proof of the fact that \(2^{n} \le {{\mathrm{lcm}}}\{1, 2, 3, \dots , (n+1)\}\). This result has a standard proof via an integral, but our proof is purely number theoretic, requiring little more than list inductions. The proof is based on manipulations of a variant of Leibniz’s Harmonic Triangle, itself a relative of Pascal’s better-known Triangle.

NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.

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Notes

  1. 1.

    We use here since we allow .

  2. 2.

    This is illustrated in Fig. 2 from the middle (step 4) to the last (step 7).

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Author information

Authors and Affiliations

  1. Australian National University, Canberra, Australia

    Hing-Lun Chan

  2. Canberra Research Laboratory, NICTA / Data61, Australian National University, Canberra, Australia

    Michael Norrish

Authors
  1. Hing-Lun Chan
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  2. Michael Norrish
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Correspondence to Hing-Lun Chan .

Editor information

Editors and Affiliations

  1. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Jasmin Christian Blanchette

  2. Inria Nancy – Grand Est, Villers-lès-Nancy, France

    Stephan Merz

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© 2016 Springer International Publishing Switzerland

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Chan, HL., Norrish, M. (2016). Proof Pearl: Bounding Least Common Multiples with Triangles. In: Blanchette, J., Merz, S. (eds) Interactive Theorem Proving. ITP 2016. Lecture Notes in Computer Science(), vol 9807. Springer, Cham. https://doi.org/10.1007/978-3-319-43144-4_9

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  • DOI: https://doi.org/10.1007/978-3-319-43144-4_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-43143-7

  • Online ISBN: 978-3-319-43144-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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