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

$x=0.179487179487179487\ldots=0.\overline{179487} \mbox{ has }4x=0.\overline{717948}=\frac{7.\overline{179487}}{10}.$

Thus

$4x=\frac{7+x}{10} \mbox{ so } x=\frac{7}{39}.$

In general, an n-parasitic number can be found as follows. Pick a one digit integer k such that kn, and take the period of the repeating decimal k/(10n−1). This will be $\frac{k}{10n-1}(10^m-1)$ 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

$\frac{1}{19}=0.\overline{052631578947368421}.$

So that for 2/19 is double that:

$\frac{2}{19}=0.\overline{105263157894736842}.$

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:

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 1016949152542372881355932203389830508474576271186440677966 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 kn 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.