# 142857 - A remarkable number

Surprising properties of cyclic numbers.
mathematics
number theory
recreational
Published

June 17, 2020

## Cyclic Numbers

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

$142857×1=142857 \\ 142857×2=285714 \\ 142857×3=428571 \\ 142857×4=571428 \\ 142857×5=714285 \\ 142857×6=857142 \\$

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?

$0,588,235, 294,117,647$

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.