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Proof Pearl: Bounding Least Common Multiples with Triangles

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Abstract

We present a proof of the fact that for \(n \ge 0\). This result has a standard proof via an integral, but our proof is purely number-theoretic, requiring little more than inductions based on lists. The almost-pictorial proof is based on manipulations of a variant of Leibniz’s harmonic triangle, itself a relative of Pascal’s better-known Triangle.

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Notes

  1. The AKS paper [1] cites Nair’s tighter LCM lower bound (Theorem 2), but as our forthcoming AKS formalization paper shows, the weaker LCM lower bound of Theorem 1 suffices.

  2. The pattern of binomial expansion for successive powers is known from antiquity, as recorded in mathematical treatises from China, Japan, India, as well as Arabic countries. For a full history of the arithmetic triangle, see Edwards [9].

  3. Note that for integer arithmetic,

  4. Another representation is based on sets. The set elements are indexed by natural numbers, so that elements can be swapped. This approach was shown to us (personal communication) by Laurent Théry. Our source script file includes a proof of Theorem 1 based on this alternative approach.

  5. For any number

  6. Here, all fractional forms are integers, as the numerator is divisible by the denominator.

  7. This is illustrated in Fig. 5 from the middle (Step 4) to the last (Step 7).

  8. A “sound wave” which is noisy at prime number times but quiet at other times, as described by Terence Tao [20].

  9. Refer to our proof script for a formalization of this recurrence formula, which requires some effort.

References

  1. Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Ann. Math. 160(2), 781–793 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asperti, A., Ricciotti, W.: About the formalization of some results by Chebyshev in number theory. In: Types for Proofs and Programs, International Conference, TYPES 2008, Torino, Italy, March 26–29, 2008, Revised Selected Papers, pp. 19–31 (2008)

  3. Avigad, J., Donnelly, K., Gray, D., Raff, P.: A formally verified proof of the prime number theorem. ACM Trans. Comput. Logic 9(1), 2 (2007). doi:10.1145/1297658.1297660

    Article  MathSciNet  MATH  Google Scholar 

  4. Ayoub, A.B.: The harmonic triangle and the beta function. Math. Mag. 60(4), 223–225 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bicknell-Johnson, M.: Diagonal sums in the harmonic triangle. Fibonacci Quart. 19(3), 196–199 (1981)

    MathSciNet  MATH  Google Scholar 

  6. Chan, H.-L., Norrish, M.: Mechanisation of AKS Algorithm: Part 1–the main theorem. In: Urban, C., Zhang, X. (eds.) Interactive Theorem Proving, ITP 2015, Number 9236 in LNCS, pp. 117–136. Springer, Berlin (2015)

    Google Scholar 

  7. Chan, H.-L., Norrish, M.: Proof pearl: bounding least common multiples with triangles. In: Blanchette, J.C., Merz, S. (eds.) Interactive Theorem Proving, ITP 2016, Number 9807 in LNCS, pp. 140–150. Springer, Berlin (2016)

    Google Scholar 

  8. Chyzak, F., Mahboubi, A., Sibut-Pinote, T., Tassi, E.: A computer-algebra-based formal proof of the irrationality of \(\zeta \)(3). In: Klein, G., Gamboa, R. (eds) Interactive Theorem Proving: 5th International Conference, ITP 2014, Held as Part of the Vienna Summer of Logic, VSL 2014, Vienna, Austria, July 14–17, 2014. Proceedings, pp. 160–176. Springer, Cham (2014)

  9. Edwards, A.W.F.: Pascal’s Arithmetical Triangle: The Story of a Mathematical Idea. Johns Hopkins University Press, Baltimore (2002)

    MATH  Google Scholar 

  10. Esteve, M.R.M., Delshams, A.: Euler’s beta function in Pietro Mengoli’s works. Arch. Hist. Exact Sci. 63(3), 325–356 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Farhi, B.: An identity involving the least common multiple of binomial coefficients and its application. Am. Math. Month. 116(9), 836–839 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fine, B., Rosenberger, G.: An epic drama: the development of the prime number theorem. Sci. Ser. A Math. Sci. 20:1–26 (2010). http://www.mat.utfsm.cl/scientia/

  13. Hanson, D.: On the product of the primes. Can. Math. Bull. 15(1), 33–37 (1972)

    Article  MATH  Google Scholar 

  14. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 6th edn. Oxford University Press, Oxford (2008). ISBN: 9780199219865

    MATH  Google Scholar 

  15. Harrison, J.: Formalizing an analytic proof of the Prime Number Theorem (dedicated to Mike Gordon on the occasion of his 60th birthday). J. Autom. Reason. 43, 243–261 (2009)

    Article  Google Scholar 

  16. Hong, S., Nair’s, F.: Identities involving the least common multiple of binomial coefficients are equivalent (2009). http://arxiv.org/pdf/0907.3401

  17. Grigory, M.: Answer to: Is there a Direct Proof of this LCM identity?. Question 1442 on Math Stack Exchange (2010). http://math.stackexchange.com/questions/1442/

  18. Nair, M.: On Chebyshev-type inequalities for primes. Am. Math. Month. 89(2), 126–129 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nicolas, A.: Answer to: Reason for LCM of all numbers from \(1 .. n\) equals roughly \(e^n\)?. Question 1111334 on Math Stack Exchange (2015). http://math.stackexchange.com/questions/1111334/

  20. Tao, T.: Structure and randomness in the prime numbers. A talk delivered at the Science colloquium (2007). https://www.math.ucla.edu/~tao/preprints/Slides/primes.pdf

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Authors and Affiliations

  1. Data61, CSIRO, Canberra, Australia

    Michael Norrish

  2. Australian National University, Canberra, Australia

    Hing-Lun Chan & Michael Norrish

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  1. Hing-Lun Chan
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Correspondence to Hing-Lun Chan.

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Chan, HL., Norrish, M. Proof Pearl: Bounding Least Common Multiples with Triangles. J Autom Reasoning 62, 171–192 (2019). https://doi.org/10.1007/s10817-017-9438-0

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