Narcissistic number
Encyclopedia
In recreational
number theory
, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) is a number that is the sum of its own digits each raised to the power of the number of digits. This definition depends on the base b of the number system used, e.g. b = 10 for the decimal system
or b = 2 for the binary system
.
The definition of a narcissistic number relies on the decimal representation n = dkdk-1...d1d0 of a natural number
n, e.g.
with k digits di satisfying 0 ≤ di ≤ 9. Such a number n is called narcissistic if it satisfies the condition
For example the 3-digit decimal number 153 is a narcissistic number because 153 = 13 + 53 + 33.
Narcissistic numbers can also be defined with respect to numeral system
s with a base b other than b = 10. The base-b representation of a natural number n is defined by
where the base-b digits di satisfy the condition 0 ≤ di ≤ b-1.
For example the (decimal) number 17 is a narcissistic number with respect to the numeral system with base b = 3. Its three base-3 digits are 122, because 17 = 1·32 + 2·3 + 2 , and it satisfies the equation 17 = 13 + 23 + 23.
If the constraint that the power must equal the number of digits is dropped, so that for some m possibly different from k it happens that
then n is called a perfect digital invariant or PDI. For example, the decimal number 4150 has four decimal digits and is the sum of the fifth powers of its decimal digits
so it is a perfect digital invariant but not a narcissistic number.
In "A Mathematician's Apology
", G. H. Hardy
wrote:
0
, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153
, 370, 371, 407, 1634, 8208, 9474 ...
The sequence of "base 3" narcissistic numbers starts:
0, 1, 2, 12, 122
The sequence of "base 4" narcissistic numbers starts:
0, 1, 2, 3, 313
The number of narcissistic numbers in a given base is finite, since the maximum possible sum of the kth powers of a k digit number in base b is
and if k is large enough then
in which case no base b narcissistic number can have k or more digits.
There are 88 narcissistic numbers in base 10, of which the largest is
with 39 digits.
Unlike narcissistic numbers, no upper bound can be determined for the size of PDIs in a given base, and it is not currently known whether or not the number of PDIs for an arbitrary base is finite or infinite.
where di are the digits of n in some base.
Recreational mathematics
Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...
number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a narcissistic number (also known as a pluperfect digital invariant (PPDI), an Armstrong number (after Michael F. Armstrong) or a plus perfect number) is a number that is the sum of its own digits each raised to the power of the number of digits. This definition depends on the base b of the number system used, e.g. b = 10 for the decimal system
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
or b = 2 for the binary system
Binary numeral system
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
.
The definition of a narcissistic number relies on the decimal representation n = dkdk-1...d1d0 of a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
n, e.g.
- n = dk·10k-1 + dk-1·10k-2 + ... + d2·10 + d1,
with k digits di satisfying 0 ≤ di ≤ 9. Such a number n is called narcissistic if it satisfies the condition
- n = dkk + dk-1k + ... + d2k + d1k.
For example the 3-digit decimal number 153 is a narcissistic number because 153 = 13 + 53 + 33.
Narcissistic numbers can also be defined with respect to numeral system
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
s with a base b other than b = 10. The base-b representation of a natural number n is defined by
- n = dkbk-1 + dk-1bk-2 + ... + d2b + d1,
where the base-b digits di satisfy the condition 0 ≤ di ≤ b-1.
For example the (decimal) number 17 is a narcissistic number with respect to the numeral system with base b = 3. Its three base-3 digits are 122, because 17 = 1·32 + 2·3 + 2 , and it satisfies the equation 17 = 13 + 23 + 23.
If the constraint that the power must equal the number of digits is dropped, so that for some m possibly different from k it happens that
- n = dkm + dk-1m + ... + d2m + d1m,
then n is called a perfect digital invariant or PDI. For example, the decimal number 4150 has four decimal digits and is the sum of the fifth powers of its decimal digits
- 4150 = 45 + 15 + 55 + 05,
so it is a perfect digital invariant but not a narcissistic number.
In "A Mathematician's Apology
A Mathematician's Apology
A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy. It concerns the aesthetics of mathematics with some personal content, and gives the layman an insight into the mind of a working mathematician.-Summary:...
", G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
wrote:
- There are just four numbers, after unity, which are the sums of the cubes of their digits:
.
- These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to the mathematician.
Narcissistic numbers in various bases
The sequence of "base 10" narcissistic numbers starts:0
0 (number)
0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...
, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153
153 (number)
One hundred fifty-three is the natural number following one hundred fifty-two and preceding one hundred fifty-four.-Mathematical properties:...
, 370, 371, 407, 1634, 8208, 9474 ...
The sequence of "base 3" narcissistic numbers starts:
0, 1, 2, 12, 122
17 (number)
17 is the natural number following 16 and preceding 18. It is prime.In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70...
The sequence of "base 4" narcissistic numbers starts:
0, 1, 2, 3, 313
55 (number)
55 is the natural number following 54 and preceding 56.-Albania:*Gazeta 55, a newspaper*Constitution law 55, a law during Communist Albania.-Mathematics:...
The number of narcissistic numbers in a given base is finite, since the maximum possible sum of the kth powers of a k digit number in base b is
and if k is large enough then
in which case no base b narcissistic number can have k or more digits.
There are 88 narcissistic numbers in base 10, of which the largest is
- 115,132,219,018,763,992,565,095,597,973,971,522,401
with 39 digits.
Unlike narcissistic numbers, no upper bound can be determined for the size of PDIs in a given base, and it is not currently known whether or not the number of PDIs for an arbitrary base is finite or infinite.
Related concepts
The term "narcissistic number" is sometimes used in a wider sense to mean a number that is equal to any mathematical manipulation of its own digits. With this wider definition narcisstic numbers include:- Constant base numbers :
for some m. - Perfect digit-to-digit invariants :
- Ascending power numbers :
- Friedman numberFriedman numberA Friedman number is an integer which, in a given base, is the result of an expression using all its own digits in combination with any of the four basic arithmetic operators and sometimes exponentiation. For example, 347 is a Friedman number since 347 = 73 + 4...
s . - Sum-product numberSum-product numberA sum-product number is an integer that in a given base is equal to the sum of its digits times the product of its digits. Or, to put it algebraically, given an integer n that is l digits long in base b , ifn = then n is a sum-product number in base b...
s :
- Dudeney numberDudeney numberA Dudeney number is a positive integer that is a perfect cube such that the sum of its decimal digits is equal to the cube root of the number. There are exactly six such integers : 1 = 1 x 1 x 1 ; 1 = 1 512 = 8 x 8 x 8 ; 8 = 5 + 1 + 2...
s :
- FactorionFactorionA factorion is a natural number that equals the sum of the factorials of its decimal digits. For example, 145 is a factorion because 1! + 4! + 5! = 1 + 24 + 120 = 145.There are just four factorions and they are 1, 2, 145 and 40585 .-Upper bound:...
s :
where di are the digits of n in some base.