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David Terr
Ph.D. Math, UC Berkeley

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5.3.8. The Smallest Cyclic Number (142857)

Would you like to see a neat number trick? Take the number 142857 and multiply it by any number from 2 to 6. The result always has the same digits in the same order, if we say the first digit comes after the last. For instance, we have 142857 * 2 = 285714. Note that in this number, just like in 142857, the digit 1 is followed by 4, which is followed by 2, which is followed by 8, which is followed by 5, which is followed by 7, which is followed again by 1. Such an arrangement of digits is called a cyclic permutation. For this reason, the number 142857 is called a cyclic number. The other products, with the same property, are 3 * 142857 = 428571, 4 * 142857 = 571428, 5 * 142857 = 714285, and 6 * 142857 = 857142. What about 7 * 142857? If you perform this multiplication, you'll get another surprise, namely 142857 * 7 = 999999! Although this is no longer a cyclic permutation of the original number, its digits have an obvious pattern - it is a 6-digit repdigit and one less than a million.

For convenience, we provide the following table, showing the multiples k*n of n = 142857 for k from 1 to 7.

k k*n
1
142857
2
285714
3
428571
4
571428
5
714285
6
857142
7
999999

Table 5.3.8.1: Small Multiples of 142857

So why does this trick work? Consider the decimal expansion of 1/7. If you do the division, you'll find that 1/7 = 0.142857, where the bar above the digits indicate that they are repeated indefinitely. If we multiply 1/7 by 10, we end up shifting all the digits in the decimal expansion one place to the left. Thus, we see that 10/7 = 1 3/7 = 1.428571, whence 3/7 = 0.428571. It is precisely for this reason that 3 * 142857 = 428571. If we multiply 3/7 by 10, we get 30/7 = 4 2/7 = 4.285714, whence 2/7 = 0.285714 and 2 * 142857 = 285714. Now if we multiply 2/7 by 10, we get 20/7 = 2 6/7 = 2.857142, whence 6 * 142857 = 857142. Multiplying 6/7 by 10, we obtain 60/7 = 8 4/7 = 8.571428, whence 4 * 142857 = 571428. Multiplying 4/7 by 10, we obtain 40/7 = 5 5/7 = 5.714285, whence 5 * 142857 = 714285. Repeating this process one more time, we have 50/7 = 7 1/7 = 7.142857, showing once again that 1/7 = 0.142857 and taking us back where we started.

Are there any other numbers with this amazing property? Yes, there are. In fact, there are infinitely many of them! Such numbers all arise from the decimal expansions of numbers of the form 1/p, where p is what is known as a cyclic prime. Below is a list of the first few cyclic primes and the corresponding cyclic numbers to which they give rise. With the exception of 142857, however, we need to be careful with these numbers. It is necessary to add a zero to the beginning of the other numbers in order for the cyclic property to work. Thus, we show the leading zero in front of these numbers.

p
n
7
142857
17
0588235294117647
19
052631578947368421
23
0434782608695652173913
29
0344827586206896551724137931
47
0212765957446808510638297872340425531914893617
59
0169491525423728813559322033898305084745762711864406779661
61
016393442622950819672131147540983606557377049180327868852459

Table 5.3.8.2: The First Eight Cyclic Numbers