142857 - A remarkable number

Cyclic Numbers

As is well known—but perhaps not as well as it should be!—the number $142,857$ has a remarkable property:

In other words,the first six multiples of $142,857$ are obtained simply by cyclically permuting its digits. The number $142,857$ is thus known as a cyclic number, that is,an $n$-digit number(possibly having some initial digits $0$) whose first $n$ multiples are given by cyclically permuting the original number in all possible ways. The existence of such a remarkable number immediately raises further questions.

First,do there exist other cyclic numbers?

The answer is yes:

is the next smallest cyclic number,having $16$ digits; multiplying it by any number from $1$ to $16$ simply cycles its digits!

The next question that arises, then, is: Are there infinitely many cyclic numbers?
This, remarkably, is an unsolved problem.

It turns out that every cyclic number in base $10$ is given by the repeating pattern in the decimal expansion of $1/p$ for a prime $p$ such that $10$ is a primitive root modulo $p$ (i.e., the first $p−1$ powers of ten—$10^0, 10^1, \dots,10^{p−2}$—have distinct remainders when divided by $p$). For example, that $142,857$ is a cyclic number is related to the fact that $1,10,10^2,10^3,10^4$, and $10^5$ all yield distinct remainders when divided by $7$,namely, $1,3,2,6,4$, and $5$,respectively.

Exactly nine primes smaller than $100$ generate cyclic numbers: $7, 17, 19, 23, 29,47, 59. 61, 97$.

Other surprising properties

1. When a cyclic number is multiplied by its generating prime, the product is always a row of $9$‘s. For instance, $142,857$ times $7$ is $999,999$. This provides another way to search for cyclics: divide a prime, $p$, into a row of $9$‘s until there is no remainder. If the quotient has $p - 1$ digits, it is a cyclic number.

2. Even less expected is the fact that every cyclic (or any of its cyclic permutations), when split in half, gives two numbers that add to a row of $9$‘s. For example, $142 + 857 = 999$. For another example, split the cyclic generated by $1/17$ into halves and add: $05882352 + 94117647 = 9999999$.

References

Martin Gardner. Cyclic numbers. In Mathematical Circus: More Puzzles, Games, Paradoxes and Other Mathematical Entertainments from Scientific American. Vintage Books, New York, 1981.