# 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.

## Contents

## Derivation[edit]

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=2
**8** - 4•
**8**7=3**48** - 4•
**48**7=1**948** - 4•
**948**7=3**7948** - 4•
**7948**7=3**17948** - 4•
**17948**7=**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*/(10*n*−1). This will be where *m* is the length of the period; i.e. the multiplicative order of 10 modulo (10*n* − 1).

For another example, if *n* = 2, then 10*n* − 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 × (10^{18} − 1)/19 = 105263157894736842.

105263157894736842 × 2 = 210526315789473684, which is the result of moving the last digit of 105263157894736842 to the front.

## Smallest n-parasitic numbers[edit]

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[edit]

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[edit]

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[edit]

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*/(10*n* - 1), also the period of the decadic integer -*n*/(10*n* - 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[edit]

## Notes[edit]

**^**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[edit]

- 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