Dudeney Number
   HOME

TheInfoList



OR:

In
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, a Dudeney number in a given number base b is a
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
equal to the perfect cube of another
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
such that the
digit sum In mathematics, the digit sum of a natural number in a given radix, number base is the sum of all its numerical digit, digits. For example, the digit sum of the decimal number 9045 would be 9 + 0 + 4 + 5 = 18. Definition Let n be a natural number. ...
of the first natural number is equal to the second. The name derives from
Henry Dudeney Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. He is known as one of the foremost creators of mathematical puzzles. Early life Dudene ...
, who noted the existence of these numbers in one of his puzzles, ''Root Extraction'', where a professor in retirement at
Colney Hatch Colney Hatch () is the historical name for a small district within the London Borough of Barnet in London, England. Colney Hatch refers to a loosely defined area centred on the northern end of Colney Hatch Lane (B550), which connects Friern ...
postulates this as a general method for root extraction.


Mathematical definition

Let n be a natural number. We define the Dudeney function for base b > 1 and
power Power may refer to: Common meanings * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power, a type of energy * Power (social and political), the ability to influence people or events Math ...
p > 0 F_ : \mathbb \rightarrow \mathbb to be the following: :F_(n) = \sum_^ \frac where k = p\left(\lfloor \log_ \rfloor + 1\right) is the p times the number of digits in the number in base b. A natural number n is a Dudeney root if it is a fixed point for F_, which occurs if F_(n) = n. The natural number m = n^p is a generalised Dudeney number, and for p = 3, the numbers are known as Dudeney numbers. 0 and 1 are trivial Dudeney numbers for all b and p, all other trivial Dudeney numbers are nontrivial trivial Dudeney numbers. For p = 3 and b = 10, there are exactly six such integers : 1, 512, 4913, 5832, 17576, 19683 A natural number n is a sociable Dudeney root if it is a
periodic point In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function (mathematics), function is a point which the system returns to after a certain number of function iterations or a certain amount of time. It ...
for F_, where F_^k(n) = n for a positive integer k, and forms a cycle of period k. A Dudeney root is a sociable Dudeney root with k = 1, and a amicable Dudeney root is a sociable Dudeney root with k = 2. Sociable Dudeney numbers and amicable Dudeney numbers are the powers of their respective roots. The number of iterations i needed for F_^(n) to reach a fixed point is the Dudeney function's
persistence Persistence or Persist may refer to: Math and computers * Image persistence, in LCD monitors * Persistence (computer science), the characteristic of data that outlives the execution of the program that created it * Persistence of a number, a ma ...
of n, and undefined if it never reaches a fixed point. It can be shown that given a number base b and power p, the maximum Dudeney root has to satisfy this bound: :n \leq (b - 1)(1 + p + \log_) = (b - 1)(1 + p + p \log_)) implying a finite number of Dudeney roots and Dudeney numbers for each order p and base b. F_ is the
digit sum In mathematics, the digit sum of a natural number in a given radix, number base is the sum of all its numerical digit, digits. For example, the digit sum of the decimal number 9045 would be 9 + 0 + 4 + 5 = 18. Definition Let n be a natural number. ...
. The only Dudeney numbers are the single-digit numbers in base b, and there are no periodic points with prime period greater than 1.


Dudeney numbers, roots, and cycles of ''F''''p'',''b'' for specific ''p'' and ''b''

All numbers are represented in base b.


Extension to negative integers

Dudeney numbers can be extended to the negative integers by use of a
signed-digit representation In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to accomplish fast addition of integers becau ...
to represent each integer.


Programming example

The example below implements the Dudeney function described in the definition above to search for Dudeney roots, numbers and cycles in
Python Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (prog ...
. def dudeneyf(x: int, p: int, b: int) -> int: """Dudeney function.""" y = pow(x, p) total = 0 while y > 0: total = total + y % b y = y // b return total def dudeneyf_cycle(x: int, p: int, b: int) -> list nt seen = [] while x not in seen: seen.append(x) x = dudeneyf(x, p, b) cycle = [] while x not in cycle: cycle.append(x) x = dudeneyf(x, p, b) return cycle


See also

*
Arithmetic dynamics Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Part of the inspiration comes from complex dynamics, the study of the Iterated function, iteration of self-maps of the complex plane or o ...
* Factorion *
Happy number In number theory, a happy number is a number which eventually reaches 1 when the number is replaced by the sum of the square of each digit. For instance, 13 is a happy number because 1^2+3^2=10, and 1^2+0^2=1. On the other hand, 4 is not a happy ...
*
Kaprekar's constant In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a four-digit random number, sorts the digits into descending and ascending order, and calculate ...
*
Kaprekar number In mathematics, a natural number in a given number base is a p-Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p digits, that add up to the original number. For example, in ...
* Meertens number * Narcissistic number *
Perfect digit-to-digit invariant In number theory, a perfect digit-to-digit invariant (PDDI; also known as a Munchausen number) is a natural number in a given number base b that is equal to the sum of its digits each raised to the power of itself. An example in base 10 is 3435, b ...
* Perfect digital invariant * Sum-product number


References

*H. E. Dudeney, ''536 Puzzles & Curious Problems'', Souvenir Press, London, 1968, p 36, #120.


External links


Generalized Dudeney Numbers

Proving There are Only Six Dudeney Numbers
{{Classes of natural numbers Arithmetic dynamics Base-dependent integer sequences