In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
, a self number or Devlali number in a given
number base
In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
that cannot be written as the sum of any other natural number
and the individual digits of
. 20 is a self number (in base 10), because no such combination can be found (all
give a result less than 20; all other
give a result greater than 20). 21 is not, because it can be written as 15 + 1 + 5 using ''n'' = 15. These numbers were first described in 1949 by the
India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and the most populous democracy in the world. Bounded by the Indian Ocean on the so ...
n
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
D. R. Kaprekar
Dattatreya Ramchandra Kaprekar ( mr, दत्तात्रेय रामचंद्र कापरेकर; 17 January 1905 – 1986) was an Indian recreational mathematician who described several classes of natural numbers incl ...
.
Definition and properties
Let
be a natural number. We define the
-self function for base
to be the following:
:
where
is the number of digits in the number in base
, and
:
is the value of each digit of the number. A natural number
is a
-self number if the
preimage
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
of
for
is the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
.
In general, for even bases, all
odd
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
numbers below the base number are self numbers, since any number below such an odd number would have to also be a 1-digit number which when added to its digit would result in an even number. For odd bases, all odd numbers are self numbers.
[Sándor & Crstici (2004) p.384]
The set of self numbers in a given base
is infinite and has a positive
asymptotic density In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the des ...
: when
is odd, this density is 1/2.
[Sándor & Crstici (2004) p.385]
Recurrent formula
The following
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
generates some
base 10
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...
self numbers:
:
(with ''C''
1 = 9)
And for
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that t ...
numbers:
:
(where ''j'' stands for the number of digits) we can generalize a recurrence relation to generate self numbers in any base ''b'':
:
in which ''C''
1 = ''b'' − 1 for even bases and ''C''
1 = ''b'' − 2 for odd bases.
The existence of these recurrence relations shows that for any base there are infinitely many self numbers.
Selfness tests
Reduction tests
Luke Pebody showed (Oct 2006) that a link can be made between the self property of a large number ''n'' and a low-order portion of that number, adjusted for digit sums:
Effective test
Kaprekar
demonstrated that:
: is self if
Where:
:
:
:
is the sum of all digits in .
:
is the number of digits in .
Self numbers in specific bases
For
base 2 self numbers, see . (written in base 10)
The first few base 10 self numbers are:
:
1,
3,
5,
7,
9,
20,
31,
42,
53,
64,
75,
86,
97,
108 108 may refer to:
* 108 (number)
* AD 108, a year
* 108 BC, a year
* 108 (artist) (born 1978), Italian street artist
* 108 (band), an American hardcore band
* 108 (emergency telephone number), an emergency telephone number in several states in Ind ...
,
110
110 may refer to:
*110 (number), natural number
*AD 110, a year
*110 BC, a year
*110 film, a cartridge-based film format used in still photography
*110 (MBTA bus), Massachusetts Bay Transportation Authority bus route
*110 (song), 2019 song by Capi ...
,
121 121 may refer to:
* 121 (number), a natural number
*AD 121, a year in the 2nd century AD
* 121 BC, a year in the 2nd century BC
* 121 (Eagle) Sqn
* 121 (MBTA bus)
* 121 (New Jersey bus)
*Road 121, see list of highways numbered 121
*Russian cruiser ...
,
132 132 may refer to:
*132 (number)
*AD 132
*132 BC
__NOTOC__
Year 132 BC was a year of the Roman calendar, pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Laenas and Rupilius (or, less frequently, year 622 ''Ab ...
,
143 143 may refer to:
*143 (number), a natural number
*AD 143, a year of the 2nd century AD
*143 BC, a year of the 2nd century BC
*143 (EP), ''143'' (EP), a 2013 EP by Tiffany Evans
*143 (album), ''143'' (album), a 2015 album by Bars and Melody
*143 (2 ...
,
154
Year 154 ( CLIV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Aurelius and Lateranus (or, less frequently, year 907 ''Ab urbe cond ...
,
165
Year 165 ( CLXV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Orfitus and Pudens (or, less frequently, year 918 ''Ab urbe condita'' ...
,
176
Year 176 ( CLXXVI) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Proculus and Aper (or, less frequently, year 929 '' Ab urbe condita'') ...
,
187
Year 187 ( CLXXXVII) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Quintius and Aelianus (or, less frequently, year 940 ''Ab urbe c ...
, 198,
209
Year 209 ( CCIX) was a common year starting on Sunday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Commodus and Lollianus (or, less frequently, year 962 '' Ab urbe cond ...
,
211
Year 211 ( CCXI) was a common year starting on Tuesday of the Julian calendar. At the time, in the Roman Empire it was known as the Year of the Consulship of Terentius and Bassus (or, less frequently, year 964 ''Ab urbe condita''). The denomin ...
,
222
__NOTOC__
Year 222 ( CCXXII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Antoninus and Severus (or, less frequently, ye ...
,
233
__NOTOC__
Year 233 ( CCXXXIII) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Claudius and Paternus (or, less frequently, year 986 ...
,
244
__NOTOC__
Year 244 (Roman numerals, CCXLIV) was a leap year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Tiberius Pollenius Armenius Peregrinus, Arm ...
,
255
__NOTOC__
Year 255 ( CCLV) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Valerianus and Gallienus (or, less frequently, year 1008 '' ...
,
266
__NOTOC__
Year 266 ( CCLXVI) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Gallienus and Sabinillus (or, less frequently, year 1019 ...
,
277
__NOTOC__
Year 277 ( CCLXXVII) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Probus and Paulinus (or, less frequently, year 1030 ''A ...
, 288, 299, 310, 312, 323, 334, 345, 356, 367, 378, 389,
400
__NOTOC__
Year 400 ( CD) was a leap year starting on Sunday (link will display the full calendar) of the Julian calendar. In the Roman Empire, it was known as the Year of the Consulship of Stilicho and Aurelianus (or, less frequently, year 11 ...
, 411, 413, 424, 435, 446, 457, 468, 479, 490, ...
In
base 12
The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead writ ...
, the self numbers are: (using inverted two and three for ten and eleven, respectively)
:1, 3, 5, 7, 9, Ɛ, 20, 31, 42, 53, 64, 75, 86, 97, ᘔ8, Ɛ9, 102, 110, 121, 132, 143, 154, 165, 176, 187, 198, 1ᘔ9, 1Ɛᘔ, 20Ɛ, 211, 222, 233, 244, 255, 266, 277, 288, 299, 2ᘔᘔ, 2ƐƐ, 310, 312, 323, 334, 345, 356, 367, 378, 389, 39ᘔ, 3ᘔƐ, 400, 411, 413, 424, 435, 446, 457, 468, 479, 48ᘔ, 49Ɛ, 4Ɛ0, 501, 512, 514, 525, 536, 547, 558, 569, 57ᘔ, 58Ɛ, 5ᘔ0, 5Ɛ1, ...
Self primes
A self prime is a self number that is
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
.
The first few self primes in base 10 are
:3, 5, 7, 31, 53, 97, 211, 233, 277, 367, 389, 457, 479, 547, 569, 613, 659, 727, 839, 883, 929, 1021, 1087, 1109, 1223, 1289, 1447, 1559, 1627, 1693, 1783, 1873, ...
The first few self primes in base 12 are: (using inverted two and three for ten and eleven, respectively)
:3, 5, 7, Ɛ, 31, 75, 255, 277, 2AA, 3BA, 435, 457, 58B, 5B1, ...
In October 2006 Luke Pebody demonstrated that the largest known
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
in base 10 that is at the same time a self number is 2
24036583−1. This is then the largest known self prime in base 10 .
Extension to negative integers
Self 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 because ...
to represent each integer.
Excerpt from the table of bases where 2007 is self
The following table was calculated in 2007.
References
* Kaprekar, D. R. ''The Mathematics of New Self-Numbers'' Devaiali (1963): 19 - 20.
*
*
*
*
{{Classes of natural numbers
Arithmetic dynamics
Base-dependent integer sequences
Inverse functions