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On some polynomial values of repdigit numbers

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Abstract

We study the equal values of repdigit numbers and the k-dimensional polygonal numbers. We state some effective finiteness theorems, and for small parameter values we completely solve the corresponding equations.

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References

  1. E. T. Avanesov, Solution of a problem on figurate numbers, Acta Arith., 12 (1966/67), 409–420.

    MathSciNet  Google Scholar 

  2. A. Baker, Transcendental number theory, Cambridge Mathematical Library, second edition, Cambridge University Press, Cambridge, 1990.

    MATH  Google Scholar 

  3. A. Baker and H. Davenport, The equations 3x 2 − 2 = y 2 and 8x 2 − 7 = z 2, Quart. J. Math. Oxford Ser. (2), 20 (1969), 129–137.

    Article  MathSciNet  Google Scholar 

  4. D. W. Ballew and R. C. Weger, Repdigit triangular numbers, J. Recreational Math., 8 (1975/76), 96–98.

    MathSciNet  Google Scholar 

  5. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I. The user language., J. Symbolic Comput., 24 (1997), 235–265.

    Article  MathSciNet  MATH  Google Scholar 

  6. B. Brindza, On a special superelliptic equation, Publ. Math. Debrecen, 39 (1991), 159–162.

    MathSciNet  MATH  Google Scholar 

  7. B. Brindza, Á. Pintér and S. Turjányi, On equal values of pyramidal and polynomial numbers, Indag. Math. (N.S.), 9 (1998), 183–185.

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Bugeaud, M. Mignotte, M. Stoll, S. Siksek and Sz. Tengely, Integral points on hyperelliptic curves, Algebra Number Theory, 2 (2008), 859–885.

    Article  MathSciNet  MATH  Google Scholar 

  9. L. E. Dickson, History of the theory of numbers, Vol. II: Diophantine analysis, Chelsea Publishing Co., New York, 1966.

    Google Scholar 

  10. H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927–930.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. L. Francis, Mathematical haystacks: another look at repunit numbers, College Math. J., 19 (1988), 240–246.

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Győry, On the Diophantine equation \(\left( {_k^n } \right) = x^l\), Acta Arith., 80 (1997), 289–295.

    MathSciNet  Google Scholar 

  13. L. Hajdu and Á. Pintér, Combinatorial diophantine equations, Publ. Math. Debrecen, 56 (2000), 391–403.

    MathSciNet  MATH  Google Scholar 

  14. J. H. Jaroma, Triangular repunit — there is but 1, Czechoslovak Math. J., 60 (2010), 1075–1077.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Keith, Repdigit polygonal numbers, J. Recreational Math., 12 (1979/80), 9–15.

    MathSciNet  MATH  Google Scholar 

  16. M. Keith, On repdigit polygonal numbers, J. Integer Sequences, 1 (1998), Article 98.1.6.

  17. P. Kiss, On the number of solutions of the Diophantine equation \(\left( {_p^x } \right) = \left( {_2^y } \right)\), Fibonacci Quart., 26 (1988), 127–130.

    MathSciNet  MATH  Google Scholar 

  18. T. Kovács, Combinatorial numbers in binary recurrences, Period. Math. Hungar., 58 (2009), 83–98.

    Article  MathSciNet  MATH  Google Scholar 

  19. Á. Pintér, A note on the Diophantine equation \(\left( {_4^x } \right) = \left( {_2^y } \right)\), Publ. Math. Debrecen, 47 (1995), 411–415.

    MathSciNet  MATH  Google Scholar 

  20. Á. Pintér and N. Varga, Resolution of a nontrivial Diophantine equation without reduction methods, Publ. Math. Debrecen, 79 (2011), 605–610.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Schinzel and R. Tijdeman, On the equation y m = P(x), Acta Arith., 31 (1976), 199–204.

    MathSciNet  Google Scholar 

  22. T. N. Shorey and R. Tijdeman, Exponential diophantine equations, Cambridge University Press, Cambridge, 1986.

    Book  MATH  Google Scholar 

  23. B. M. M. De Weger, A binomial Diophantine equation, Quart. J. Math. Oxford Ser. (2), 47 (1996), 221–231

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to T. Kovács.

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Communicated by Attila Pethő

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Kovács, T., Péter, G. & Varga, N. On some polynomial values of repdigit numbers. Period Math Hung 67, 221–230 (2013). https://doi.org/10.1007/s10998-013-1396-7

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  • DOI: https://doi.org/10.1007/s10998-013-1396-7

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