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PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE

Year 2019, Volume: 25 Issue: 25, 199 - 211, 08.01.2019

Horst Brunotte

https://doi.org/10.24330/ieja.504153

Abstract

We study perfect numbers which are repdigits in a given negative
base. It is shown that in each negative base there are at most nitely many
perfect repdigits, and that the set of all such numbers can e ectively be computed.
As an illustration we explicitly determine these numbers in bases -2
and -10.

Keywords

Perfect number repdigit

References

  • K. A. Broughan, S. G. Sanchez, and F. Luca, Perfect repdigits, Math. Comp., 82(284) (2013), 2439-2459.
  • K. A. Broughan and Q. Zhou, Odd repdigits to small bases are not perfect, Integers, 12(5) (2012), 841-858.
  • Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, Perfect powers from products of terms in Lucas sequences, J. Reine Angew. Math., 611 (2007), 109-129.
  • C. Frougny and A. C. Lai, Negative bases and automata, Discrete Math. Theor. Comput. Sci., 13(1) (2011), 75-93.
  • V. Grunwald, Intorno all'aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll'aritmetica ordinaria (decimale), Giornale di matematiche di Battaglini, 23 (1885), 203-221, 367.
  • W. Ljunggren, Some theorems on indeterminate equations of the form xn 􀀀 1=x 􀀀 1 = yq, Norsk Mat. Tidsskr., 25 (1943), 17-20.
  • F. Luca, Perfect Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo, 49(2) (2000), 313-318.
  • F. Luca, Multiply perfect numbers in Lucas sequences with odd parameters, Publ. Math. Debrecen, 58(1-2) (2001), 121-155.
  • R. B. Nelsen, Even perfect numbers end in 6 or 28, Math. Mag., 91(2) (2018), 140-141.
  • P. Pollack, Perfect numbers with identical digits, Integers, 11(4) (2011), 519- 529.
  • P. Pollack and C. Pomerance, Some problems of Erd}os on the sum-of-divisors function, Trans. Amer. Math. Soc. Ser. B, 3 (2016), 1-26.
  • I. SageMath, CoCalc Collaborative Computation Online, 2017. https://cocalc.com/.
  • H. N. Shapiro, Introduction to the Theory of Numbers, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1983.
  • T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Mathematics, 87, Cambridge University Press, Cambridge, 1986.

Year 2019, Volume: 25 Issue: 25, 199 - 211, 08.01.2019

Horst Brunotte

https://doi.org/10.24330/ieja.504153

Abstract

References

  • K. A. Broughan, S. G. Sanchez, and F. Luca, Perfect repdigits, Math. Comp., 82(284) (2013), 2439-2459.
  • K. A. Broughan and Q. Zhou, Odd repdigits to small bases are not perfect, Integers, 12(5) (2012), 841-858.
  • Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, Perfect powers from products of terms in Lucas sequences, J. Reine Angew. Math., 611 (2007), 109-129.
  • C. Frougny and A. C. Lai, Negative bases and automata, Discrete Math. Theor. Comput. Sci., 13(1) (2011), 75-93.
  • V. Grunwald, Intorno all'aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll'aritmetica ordinaria (decimale), Giornale di matematiche di Battaglini, 23 (1885), 203-221, 367.
  • W. Ljunggren, Some theorems on indeterminate equations of the form xn 􀀀 1=x 􀀀 1 = yq, Norsk Mat. Tidsskr., 25 (1943), 17-20.
  • F. Luca, Perfect Fibonacci and Lucas numbers, Rend. Circ. Mat. Palermo, 49(2) (2000), 313-318.
  • F. Luca, Multiply perfect numbers in Lucas sequences with odd parameters, Publ. Math. Debrecen, 58(1-2) (2001), 121-155.
  • R. B. Nelsen, Even perfect numbers end in 6 or 28, Math. Mag., 91(2) (2018), 140-141.
  • P. Pollack, Perfect numbers with identical digits, Integers, 11(4) (2011), 519- 529.
  • P. Pollack and C. Pomerance, Some problems of Erd}os on the sum-of-divisors function, Trans. Amer. Math. Soc. Ser. B, 3 (2016), 1-26.
  • I. SageMath, CoCalc Collaborative Computation Online, 2017. https://cocalc.com/.
  • H. N. Shapiro, Introduction to the Theory of Numbers, Pure and Applied Mathematics, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1983.
  • T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, Cambridge Tracts in Mathematics, 87, Cambridge University Press, Cambridge, 1986.
There are 14 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Horst Brunotte This is me

Publication Date January 8, 2019
Published in Issue Year 2019 Volume: 25 Issue: 25

Cite

APA Brunotte, H. (2019). PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE. International Electronic Journal of Algebra, 25(25), 199-211. https://doi.org/10.24330/ieja.504153
AMA Brunotte H. PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE. IEJA. January 2019;25(25):199-211. doi:10.24330/ieja.504153
Chicago Brunotte, Horst. “PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE”. International Electronic Journal of Algebra 25, no. 25 (January 2019): 199-211. https://doi.org/10.24330/ieja.504153.
EndNote Brunotte H (January 1, 2019) PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE. International Electronic Journal of Algebra 25 25 199–211.
IEEE H. Brunotte, “PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE”, IEJA, vol. 25, no. 25, pp. 199–211, 2019, doi: 10.24330/ieja.504153.
ISNAD Brunotte, Horst. “PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE”. International Electronic Journal of Algebra 25/25 (January 2019), 199-211. https://doi.org/10.24330/ieja.504153.
JAMA Brunotte H. PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE. IEJA. 2019;25:199–211.
MLA Brunotte, Horst. “PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE”. International Electronic Journal of Algebra, vol. 25, no. 25, 2019, pp. 199-11, doi:10.24330/ieja.504153.
Vancouver Brunotte H. PERFECT NUMBERS WITH IDENTICAL DIGITS IN NEGATIVE BASE. IEJA. 2019;25(25):199-211.