Parasitic number
Encyclopedia
An n-parasitic number is a positive natural number
which can be multiplied
by n by moving the rightmost digit
of its decimal representation
to the front. Here n is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right circular shift
by one place. For example, 4•128205=512820, so 128205 is 4-parasitic. Most authors do not allow leading zeros to be used, and this article follows that convention. So even though 4•025641=102564, the number 025641 is not 4-parasitic.
For example, for n = 4 and k = 7:
So 179487 is a 4-parasitic number with units digit 7. Others are 179487179487, 179487179487179487 etc.
Notice that the repeating decimal
Thus
In general, an n-parasitic number can be found as follows. Pick a one digit integer k such that , and take the period of the repeating decimal
k/(10n−1).
This will be
where m is the length of the period; i.e. the order
of 10 modulo
.
For another example, if n = 2, then 10n − 1 = 19 and the repeating decimal for 1/19 is
So that for 2/19 is double that:
The length m of this period is 18, the same as the order of 10 modulo 19, so = 105263157894736842.
105263157894736842 × 2 = 210526315789473684, which is the result of moving the last digit of 105263157894736842 to the front.
. They are:
Other n-parasitic integers can be built by concatenation. For example, since 179487 is a 4-parasitic number, so are 179487179487, 179487179487179487 etc.
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...
which can be multiplied
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
by n by moving the rightmost digit
Digit
Digit may refer to:* Digit , one of several most distal parts of a limb—fingers, thumbs, and toes on hands and feet* Numerical digit, as used in mathematics or computer science* Hexadecimal, representing a four-bit number...
of its decimal representation
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
to the front. Here n is itself a single-digit positive natural number. In other words, the decimal representation undergoes a right circular shift
Circular shift
In combinatorial mathematics, a circular shift is the operation of rearranging the entries in a tuple, either by moving the final entry to the first position, while shifting all other entries to the next position, or by performing the inverse operation...
by one place. For example, 4•128205=512820, so 128205 is 4-parasitic. Most authors do not allow leading zeros to be used, and this article follows that convention. So even though 4•025641=102564, the number 025641 is not 4-parasitic.
Derivation
An n-parasitic number can be derived by starting with a digit k (which should be equal to n or greater) in the rightmost (units) place, and working up one digit at a time.For example, for n = 4 and k = 7:
- 4•7=28
- 4•87=348
- 4•487=1948
- 4•9487=37948
- 4•79487=317948
- 4•179487=717948.
So 179487 is a 4-parasitic number with units digit 7. Others are 179487179487, 179487179487179487 etc.
Notice that the repeating decimal
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...
Thus
In general, an n-parasitic number can be found as follows. Pick a one digit integer k such that , and take the period of the repeating decimal
Repeating decimal
In arithmetic, a decimal representation of a real number is called a repeating decimal if at some point it becomes periodic, that is, if there is some finite sequence of digits that is repeated indefinitely...
k/(10n−1).
This will be
where m is the length of the period; i.e. the order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
of 10 modulo
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
.
For another example, if n = 2, then 10n − 1 = 19 and the repeating decimal for 1/19 is
So that for 2/19 is double that:
The length m of this period is 18, the same as the order of 10 modulo 19, so = 105263157894736842.
105263157894736842 × 2 = 210526315789473684, which is the result of moving the last digit of 105263157894736842 to the front.
Smallest n-parasitic numbers
The smallest n-parasitic numbers are also known as Dyson numbers, after a puzzle concerning these numbers posed by Freeman DysonFreeman Dyson
Freeman John Dyson FRS is a British-born American theoretical physicist and mathematician, famous for his work in quantum field theory, solid-state physics, astronomy and nuclear engineering. Dyson is a member of the Board of Sponsors of the Bulletin of the Atomic Scientists...
. They are:
| n | Smallest n-parasitic number | period of |
| 1 | 1 | 1/9 |
| 2 | 105263157894736842 | 2/19 |
| 3 | 1034482758620689655172413793 | 3/29 |
| 4 | 102564 | 4/39 |
| 5 | 142857 | 7/49=1/7 |
| 6 | 10169491525423728813559932203389830508474576271186440677966 | 6/59 |
| 7 | 1014492753623188405797 | 7/69 |
| 8 | 1012658227848 | 8/79 |
| 9 | 10112359550561797752808988764044943820224719 | 9/89 |
General note
In general, if we relax the rules to allow a leading zero, then there are 9 n-parasitic numbers for each n. Otherwise only if k ≥ n then the numbers do not start with zero and hence fit the actual definition.Other n-parasitic integers can be built by concatenation. For example, since 179487 is a 4-parasitic number, so are 179487179487, 179487179487179487 etc.