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Additive Number Theory via Automata Theory

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Abstract

We show how some problems in additive number theory can be attacked in a novel way, using techniques from the theory of finite automata. We start by recalling the relationship between first-order logic and finite automata, and use this relationship to solve several problems involving sums of numbers defined by their base-2 and Fibonacci representations. Next, we turn to harder results. Recently, Cilleruelo, Luca, & Baxter proved, for all bases b ≥ 5, that every natural number is the sum of at most 3 natural numbers whose base-b representation is a palindrome (Cilleruelo et al., Math. Comput. 87, 3023–3055, 2018). However, the cases b = 2, 3, 4 were left unresolved. We prove that every natural number is the sum of at most 4 natural numbers whose base-2 representation is a palindrome. Here the constant 4 is optimal. We obtain similar results for bases 3 and 4, thus completely resolving the problem of palindromes as an additive basis. We consider some other variations on this problem, and prove similar results. We argue that heavily case-based proofs are a good signal that a decision procedure may help to automate the proof.

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References

  1. Allouche, J.-P., Rampersad, N., Shallit, J.: Periodicity, repetitions, and orbits of an automatic sequence. Theoret. Comput. Sci. 410, 2795–2803 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aloui, K., Mauduit, C., Mkaouar, M.: Somme des chiffres et répartition dans les classes de congruence pour les palindromes ellipséphiques. Acta Math. Hung. 151, 409–455 (2017)

    Article  MATH  Google Scholar 

  3. Alur, R., Madhusudan, P.: Visibly Pushdown Languages. In: Proc. Thirty-Sixth Ann. ACM Symp. Theor. Comput. (STOC), pp. 202–211, ACM (2004)

  4. Alur, R., Madhusudan, P.: Adding nested structure to words. J. Assoc. Comput. Mach. 56 (2009), Art. 16

  5. Appel, K., Haken, W.: Every planar map is four colorable. I. Discharging Illinois J. Math. 21, 429–490 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  6. Appel, K., Haken, W., Koch, J.: Every planar map is four colorable. II. Reducibility Illinois J. Math. 21, 491–567 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ashbacher, C.: More on palindromic squares. J. Recreat. Math. 22, 133–135 (1990)

    MATH  Google Scholar 

  8. Banks, W.D.: Every natural number is the sum of forty-nine palindromes. INTEGERS — Electronic J. Combinat. Number Theory 16 (2016), #A3 (electronic)

  9. Banks, W.D., Hart, D.N., Sakata, M.: Almost all palindromes are composite. Math. Res. Lett. 11, 853–868 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Banks, W.D., Shparlinski, I.E.: Prime divisors of palindromes. Period. Math Hung. 51, 1–10 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Banks, W.D., Shparlinski, I.E.: Average value of the Euler function on binary palindromes. Bull. Pol. Acad. Sci Math. 54, 95–101 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Barnett, J.K.R.: Tables of square palindromes in bases 2 and 10. J. Recreat. Math. 23(1), 13–18 (1991)

    MATH  Google Scholar 

  13. Basić, B. : On d-digit palindromes in different bases: the number of bases is unbounded. Internat. J. Number Theory 8, 1387–1390 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Basić, B.: On “very palindromic” sequences. J. Korean Math. Soc. 52, 765–780 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bell, J., Hare, K., Shallit, J.: When is an automatic set an additive basis. Proc. Amer. Math. Soc. Ser. B 5, 50–63 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bell, J.P., Lidbetter, T.F., Shallit, J.: Additive number theory via approximation by regular languages. In: Hoshi, M., Seki, S. (eds.) DLT Vol. 11088 of Lecture Notes in Computer Science, pp 121–132. Springer, Berlin (2018)

  17. Bérczes, A., Ziegler, V.: On simultaneous palindromes. J. Combin. Number Theory 6, 37–49 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways for your Mathematical Plays, Vol. 2 Games in Particular. Academic Press, Cambridge (1982)

    MATH  Google Scholar 

  19. Berstel, J.: Axel Thue’s Papers on Repetitions in Words: a Translation. Number 20 in Publications Du Laboratoire De Combinatoire Et d’Informatique Mathématique. Université du Québec à Montréal (1995)

  20. von Braunmühl, B., Verbeek, R.: Input-driven languages are recognized in n space. In: Karpinski, M. (ed.) Foundations of Computation Theory, FCT 83, Vol. 158 of Lecture Notes in Computer Science, pp 40–51. Springer, Berlin (1983)

  21. Bruyère, V., Hansel, G.: Bertrand numeration systems and recognizability. Theoret. Comput. Sci. 181, 17–43 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Bruyère, V., Hansel, G., Michaux, C., Villemaire, R.: Logic and p-recognizable sets of integers. Bull. Belgian Math. Soc. 1, 191–238 (1994). Corrigendum, Bull. Belg. Math. Soc. 1 (1994) 577

    MathSciNet  MATH  Google Scholar 

  23. Büchi, J.R.: Weak secord-order arithmetic and finite automata. Zeitschrift fur̈ mathematische Logik und Grundlagen der Mathematik 6, 66–92 (1960). Reprinted in S. Mac Lane and D. Siefkes, eds., The Collected Works of J. Richard Buchï, Springer, 1990, pp.398–424

    Article  MATH  Google Scholar 

  24. Charlier, E., Rampersad, N., Shallit, J.: Enumeration and decidable properties of automatic sequences. Internat. J. Found. Comp. Sci. 23, 1035–1066 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Cilleruelo, J., Luca, F.: Every positive integer is a sum of three palindromes. arXiv:1602.06208v1 (2016)

  26. Cilleruelo, J., Luca, F., Baxter, L.: Every positive integer is a sum of three palindromes. Math. Comput. 87, 3023–3055 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Cilleruelo, J., Luca, F., Shparlinski, I.E.: Power values of palindromes. J Combin. Number Theory 1, 101–107 (2009)

    MathSciNet  MATH  Google Scholar 

  28. Cilleruelo, J., Tesoro, R., Luca, F.: Palindromes in linear recurrence sequences. Monatsh. Math. 171, 433–442 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Col, S.: Palindromes dans les progressions arithmétiques. Acta Arith. 137, 1–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Du, C.F., Mousavi, H., Rowland, E., Schaeffer, L.: J. Shallit. Decision algorithms for Fibonacci-automatic words II: related sequences and avoidability. Theoret. Comput. Sci. 657, 146–162 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  31. Du, C.F., Mousavi, H., Schaeffer, L., Shallit, J.: Decision algorithms for Fibonacci-automatic words III: enumeration and abelian properties. Internat. J. Found. Comp. Sci. 27, 943–963 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dymond, P.W.: Input-driven languages are in n depth. Inform. Process. Lett. 26, 247–250 (1988)

    Article  MathSciNet  Google Scholar 

  33. Goč, D., Henshall, D., Shallit, J.: Automatic theorem-proving in combinatorics on words. In: Moreira, N., Reis, R. (eds.) CIAA 2012, Vol. 7381 of Lecture Notes in Computer Science, pp 180–191. Springer, Berlin (2012)

  34. Goč, D., Mousavi, H., Shallit, J.: On the number of unbordered factors. In: Dediu, A.-H., Martin-Vide, C., Truthe, B. (eds.) LATA 2013, Vol. 7810 of Lecture Notes in Computer Science, pp 299–310. Springer, Berlin (2013)

  35. Goč, D., Saari, K., Shallit, J.: Primitive words and Lyndon words in automatic and linearly recurrent sequences. In: Dediu, A.-H., Martin-Vide, C., Truthe, B. (eds.) LATA 2013, Primitive Vol. 7810 of Lecture Notes in Computer Science, pp 311–322. Springers, Berlin (2013)

  36. Goč, D., Schaeffer, L., Shallit, J.: The subword complexity of k-automatic sequences is k-synchronized. In: Béal, M.-P., Carton, O. (eds.) DLT 2013, Vol. 7907 of Lecture Notes in Computer Science, pp 252–263. Springer, Berlin (2013)

  37. Goins, E.H.: Palindromes in different bases: a conjecture of J. Ernest Wilkins. INTEGERS — Electronic J. Combinat. Number Theory 9, 725–734 (2009). Paper #A55, (electronic)

    MathSciNet  MATH  Google Scholar 

  38. Han, Y.-S., Salomaa, K.: Nondeterministic state complexity of nested word automata. Theoret. Comput. Sci. 410, 2961–2971 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Haque, S., Shallit, J.: Discriminators and k-regular sequences. INTEGERS — Electronic J. Combinat. Number Theory 16, A76 (2016). (electronic)

    MathSciNet  MATH  Google Scholar 

  40. Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1985)

  41. Harminc, M., Soták, R.: Palindromic numbers in arithmetic progressions. Fibonacci Quart. 36, 259–262 (1998)

    MathSciNet  MATH  Google Scholar 

  42. Heizmann, M., Hoenicke, J., Podelski, A.: Software model checking for people who love automata. In: Sharygina, N., Veith, H. (eds.) Computer Aided Verification — 25th International Conference, CAV 2013, vol. 8044 of Lecture Notes in Computer Science, pp 36–52. Springer, Berlin (2013)

  43. Heizmann, M., Dietsch, D., Greitschus, M., Leike, J., Musa, B., Schätzle, C., Podelski, A.: Ultimate automizer with two-track proofs. In: Chechik, M., Raskin, J.-F. (eds.) Tools and Algorithms for the Construction and Analysis of Systems — 22nd International Conference, TACAS 2016, vol. 9636 of Lecture Notes in Computer Science, pp 950–953. Springer, Berlin (2016)

  44. Hernández, S., Luca, F.: Palindromic powers. Rev. Colombiana Mat. 40, 81–86 (2006)

    MathSciNet  MATH  Google Scholar 

  45. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Boston (1979)

    MATH  Google Scholar 

  46. Kane, D.M., Sanna, C., Shallit, J.: Waring’s Theorem for Binary Powers. arXiv:1801.04483. To appear, Combinatorica (2018)

  47. Keith, M.: Classification and enumeration of palindromic squares. J. Recreat. Math. 22(2), 124–132 (1990)

    MATH  Google Scholar 

  48. Kidder, P., Kwong, H.: Remarks on integer palindromes. J. Recreat. Math. 36, 299–306 (2007)

    MathSciNet  Google Scholar 

  49. Korec, I.: Palindromic squares for various number system bases. Math. Slovaca 41, 261–276 (1991)

    MathSciNet  MATH  Google Scholar 

  50. Kresová, H., Sǎlát, T.: On palindromic numbers. Acta Math. Univ Comenian. 42(/43), 293–298 (1984)

    MathSciNet  MATH  Google Scholar 

  51. La Torre, S., Napoli, M., Parente, M.: On the membership problem for visibly pushdown languages. In: Graf, S., Zhang, W. (eds.) ATVA 2006, vol. 4218 of Lecture Notes in Computer Science, pp. 96–109, Springer (2006)

  52. Lekkerkerker, C.G.: Voorstelling van natuurlijke getallen door een som van getallen van Fibonacci. Simon Stevin 29, 190–195 (1952)

    MathSciNet  Google Scholar 

  53. Luca, F.: Palindromes in Lucas sequences. Monatsh. Math. 138, 209–223 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  54. Luca, F., Togbé, A.: On binary palindromes of the form 10n1. C. R. Acad. Sci. Paris 346, 487–489 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  55. Luca, F., Young, P.T.: On the Binary Expansion of the Odd Catalan Numbers. In: Proc. 14Th Int. Conf. on Fibonacci Numbers and Their Applications, pp. 185–190. Soc. Mat. Mexicana (2011)

  56. Madhusudan, P., Nowotka, D., Rajasekaran, A., Shallit, J.: Lagrange’s theorem for binary squares. In: Potapov, I., Spirakis, P., Worrell, J. (eds.) 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), Leibniz International Proceedings in Informatics, pp. 18:1–18:14. Schloss Dagstuhl—Leibniz-Zentrum für Informatik, Germany (2018)

  57. McCune, W.: Solution of the Robbins problem. J. Autom. Reason. 19(3), 263–276 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  58. Mehlhorn, K.: Pebbling Mountain Ranges and Its Application of DCFL-Recognition. In: Proc. 7Th Int’l Conf. on Automata, Languages, and Programming (ICALP), Vol. 85 of Lecture Notes in Computer Science, pp. 422–435. Springer (1980)

  59. Mousavi, H.: Automatic theorem proving in Walnut. arXiv:1603.06017 (2016)

  60. Mousavi, H., Schaeffer, L., Shallit, J.: Decision algorithms for Fibonacci-automatic words, I Basic results. RAIRO Inform. Théor. App. 50, 39–66 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  61. Nathanson, M.B.: Additive Number Theory: The Classical Bases. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  62. Ogden, W.: A helpful result for proving inherent ambiguity. Math. Systems Theory 2, 191–194 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  63. Okhotin, A., Salomaa, K.: State complexity of operations on input-driven pushdown automata. J Comput. System Sci. 86, 207–228 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  64. Piao, X., Salomaa, K.: Operational state complexity of nested word automata. Theoret. Comput. Sci. 410, 3290–3302 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  65. Pollack, P.: Palindromic sums of proper divisors. INTEGERS — Electronic J. Combinat. Number Theory 15A, Paper #A13 (electronic) (2015)

    MathSciNet  MATH  Google Scholar 

  66. Rajasekaran, A.: Using automata theory to solve problems in additive number theory. Master’s thesis, School of Computer Science, University of Waterloo, 2018. Available at https://uwspace.uwaterloo.ca/handle/10012/13202

  67. Rajasekaran, A., Smith, T., Shallit, J.: Sums of palindromes: an approach via automata. In: Niedermeier, R., Vallée, B. (eds.) 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018), Leibniz International Proceedings in Informatics, pp. 54:1–54:12. Schloss Dagstuhl — Leibniz-Zentrum fur̈ Informatik (2018)

  68. Salomaa, K.: Limitations of lower bound methods for deterministic nested word automata. Inform. Comput. 209, 580–589 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  69. Shallit, J.: Decidability and enumeration for automatic sequences: a survey. In: Bulatov, A.A., Shur, A.M. (eds.) CSR 2013, vol. 7913 of Lecture Notes in Computer Science, pp 49–63. Springer, Berlin (2013)

  70. Shorey, T.N.: On the equation zq = (xn − 1)/(x − 1). Indag. Math. 48, 345–351 (1986)

    Article  MathSciNet  Google Scholar 

  71. Sigg, M.: On a conjecture of John Hoffman regarding sums of palindromic numbers. arXiv:1510.07507 (2015)

  72. Simmons, G.J.: On palindromic squares of non-palindromic numbers. J. Recreat. Math. 5(1), 11–19 (1972)

    MathSciNet  Google Scholar 

  73. Sloane, N.J.A., et al.: The on-line encyclopedia of integer sequences Available at https://oeis.org (2019)

  74. Thue, A.: ÜBer die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske vid. Selsk. Skr. Mat. Nat. Kl. 1, 1–67 (1912). Reprinted in Selected Mathematical Papers of Axel Thue, T. Nagell, editor, Universitetsforlaget, Oslo, 1977, 413–478

    MATH  Google Scholar 

  75. Trigg, C.: Infinite sequences of palindromic triangular numbers. Fibonacci Quart. 12, 209–212 (1974)

    MathSciNet  MATH  Google Scholar 

  76. Tymoczko, T.: The four-color problem and its philosophical significance. J. Philosophy 76(2), 57–83 (1979)

    Article  Google Scholar 

  77. Vaughan, R.C., Wooley, T.: Number Theory for the Millennium. III, pp. 301–340. A. K. Peters. In: Bennett, M.A., Berndt, B.C., Boston, N., Diamond, H.G., Hildebrand, A.J., Philipp, W. (eds.) (2002)

  78. Yates, S.: The mystique of repunits. Math. Mag. 51, 22–28 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  79. Zeckendorf, E.: Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas. Bull. Soc. Roy. Liége 41, 179–182 (1972)

    MathSciNet  MATH  Google Scholar 

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Acknowledgments

We thank Dirk Nowotka, Parthasarathy Madhusudan, and Jean-Paul Allouche for helpful discussions. We thank the creators of the ULTIMATE automaton library for their assistance. We are also grateful to the referees for reading our paper with care and making several helpful suggestions.

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  1. School of Computer Science, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Aayush Rajasekaran, Jeffrey Shallit & Tim Smith

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  1. Aayush Rajasekaran
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Rajasekaran, A., Shallit, J. & Smith, T. Additive Number Theory via Automata Theory. Theory Comput Syst 64, 542–567 (2020). https://doi.org/10.1007/s00224-019-09929-9

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