Interesting Esoterica

On Kaprekar's Junction Numbers

Article by Max A. Alekseyev and Donovan Johnson and N. J. A. Sloane
  • Published in 2021
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A base b junction number u has the property that there are at least two ways to write it as u = v + s(v), where s(v) is the sum of the digits in the expansion of the number v in base b. For the base 10 case, Kaprekar in the 1950's and 1960's studied the problem of finding K(n), the smallest u such that the equation u=v+s(v) has exactly n solutions. He gave the values K(2)=101, K(3)=10^13+1, and conjectured that K(4)=10^24+102. In 1966 Narasinga Rao gave the upper bound 10^1111111111124+102 for K(5), as well as upper bounds for K(6), K(7), K(8), and K(16). In the present work, we derive a set of recurrences, which determine K(n) for any base b and in particular imply that these conjectured values of K(n) are correct. The key to our approach is an apparently new recurrence for F(u), the number of solutions to u=v+s(v). We have applied our method to compute K(n) for n <= 16 and bases b <= 10. These sequences grow extremely rapidly. Rather surprisingly, the solution to the base 5 problem is determined by the classical Thue-Morse sequence. For a fixed b, it appears that K(n) grows as a tower of height about log_2(n).

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BibTeX entry

@article{OnKaprekarsJunctionNumbers,
	title = {On Kaprekar's Junction Numbers},
	abstract = {A base b junction number u has the property that there are at least two ways
to write it as u = v + s(v), where s(v) is the sum of the digits in the
expansion of the number v in base b. For the base 10 case, Kaprekar in the
1950's and 1960's studied the problem of finding K(n), the smallest u such that
the equation u=v+s(v) has exactly n solutions. He gave the values K(2)=101,
K(3)=10^13+1, and conjectured that K(4)=10^24+102. In 1966 Narasinga Rao gave
the upper bound 10^1111111111124+102 for K(5), as well as upper bounds for
K(6), K(7), K(8), and K(16). In the present work, we derive a set of
recurrences, which determine K(n) for any base b and in particular imply that
these conjectured values of K(n) are correct. The key to our approach is an
apparently new recurrence for F(u), the number of solutions to u=v+s(v). We
have applied our method to compute K(n) for n <= 16 and bases b <= 10. These
sequences grow extremely rapidly. Rather surprisingly, the solution to the base
5 problem is determined by the classical Thue-Morse sequence. For a fixed b, it
appears that K(n) grows as a tower of height about log{\_}2(n).},
	url = {http://arxiv.org/abs/2112.14365v1 http://arxiv.org/pdf/2112.14365v1},
	year = 2021,
	author = {Max A. Alekseyev and Donovan Johnson and N. J. A. Sloane},
	comment = {},
	urldate = {2022-01-10},
	archivePrefix = {arXiv},
	eprint = {2112.14365},
	primaryClass = {math.NT},
	collections = {fun-maths-facts,integerology}
}