Parasitic number
An n-parasitic number (in base 10) 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.
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
Thus
In general, an n-parasitic number can be found as follows. Pick a one digit integer k such that k ≥ n, 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 multiplicative order of 10 modulo (10n − 1).
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 2 × (1018 − 1)/19 = 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 Dyson.[1][2][3] They are: (leading zeros are not allowed) (sequence A092697 in the OEIS)
n | Smallest n-parasitic number | Digits | Period of |
---|---|---|---|
1 | 1 | 1 | 1/9 |
2 | 105263157894736842 | 18 | 2/19 |
3 | 1034482758620689655172413793 | 28 | 3/29 |
4 | 102564 | 6 | 4/39 |
5 | 142857 | 6 | 7/49 = 1/7 |
6 | 1016949152542372881355932203389830508474576271186440677966 | 58 | 6/59 |
7 | 1014492753623188405797 | 22 | 7/69 |
8 | 1012658227848 | 13 | 8/79 |
9 | 10112359550561797752808988764044943820224719 | 44 | 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.
Other bases
In duodecimal system, the smallest n-parasitic numbers are: (using inverted two and three for ten and eleven, respectively) (leading zeros are not allowed)
n | Smallest n-parasitic number | Digits | Period of |
---|---|---|---|
1 | 1 | 1 | 1/Ɛ |
2 | 10631694842 | Ɛ | 2/1Ɛ |
3 | 2497 | 4 | 7/2Ɛ = 1/5 |
4 | 10309236ᘔ88206164719544 | 1Ɛ | 4/3Ɛ |
5 | 1025355ᘔ9433073ᘔ458409919Ɛ715 | 25 | 5/4Ɛ |
6 | 1020408142854ᘔ997732650ᘔ18346916306 | 2Ɛ | 6/5Ɛ |
7 | 101899Ɛ864406Ɛ33ᘔᘔ15423913745949305255Ɛ17 | 35 | 7/6Ɛ |
8 | 131ᘔ8ᘔ | 6 | ᘔ/7Ɛ = 2/17 |
9 | 101419648634459Ɛ9384Ɛ26Ɛ533040547216ᘔ1155Ɛ3Ɛ12978ᘔ399 | 45 | 9/8Ɛ |
ᘔ | 14Ɛ36429ᘔ7085792 | 14 | 12/9Ɛ = 2/15 |
Ɛ | 1011235930336ᘔ53909ᘔ873Ɛ325819Ɛ9975055Ɛ54ᘔ3145ᘔ42694157078404491Ɛ | 55 | Ɛ/ᘔƐ |
Strict definition
In strict definition, least number m beginning with 1 such that the quotient m/n is obtained merely by shifting the leftmost digit 1 of m to the right end are
- 1, 105263157894736842, 1034482758620689655172413793, 102564, 102040816326530612244897959183673469387755, 1016949152542372881355932203389830508474576271186440677966, 1014492753623188405797, 1012658227848, 10112359550561797752808988764044943820224719, 10, 100917431192660550458715596330275229357798165137614678899082568807339449541284403669724770642201834862385321, 100840336134453781512605042016806722689075630252, ... (sequence A128857 in the OEIS)
They are the period of n/(10n - 1), also the period of the decadic integer -n/(10n - 1).
Number of digits of them are
- 1, 18, 28, 6, 42, 58, 22, 13, 44, 2, 108, 48, 21, 46, 148, 13, 78, 178, 6, 99, 18, 8, 228, 7, 41, 6, 268, 15, 272, 66, 34, 28, 138, 112, 116, 179, 5, 378, 388, 18, 204, 418, 6, 219, 32, 48, 66, 239, 81, 498, ... (sequence A128858 in the OEIS)
See also
Notes
- ↑ Dawidoff, Nicholas (March 25, 2009), "The Civil Heretic", New York Times Magazine.
- ↑ Tierney, John (April 6, 2009), "Freeman Dyson's 4th-Grade Math Puzzle", New York Times.
- ↑ Tierney, John (April 13, 2009), "Prize for Dyson Puzzle", New York Times.
References
- C. A. Pickover, Wonders of Numbers, Chapter 28, Oxford University Press UK, 2000.
- Sequence A092697 in the On-Line Encyclopedia of Integer Sequences.
- Bernstein, Leon (1968), "Multiplicative twins and primitive roots", Mathematische Zeitschrift, 105: 49–58, doi:10.1007/BF01135448, MR 0225709