The Mathematical Carousel

Let’s start with the number in focus: 142857.

First, I'll organize the digits of this number in ascending order (note that it has six digits):

1, 2, 4, 5, 7, 8

Now, returning to the original number, let’s multiply it by 1:

142857 * 1 = 142857

This is straightforward; multiplying any number by one yields the same number. However, it’s important to note that this result begins with the smallest digit, which is 1.

Next, let’s multiply the number by 2:

142857 * 2 = 285714

At first glance, this seems non-trivial, but upon closer inspection, it’s simply a rearrangement of 142857, starting with the second smallest digit.

Continuing on, what happens when we multiply 142857 by 3?

142857 * 3 = 428571

Again, we see another permutation of 142857, this time starting with the third smallest digit (4). By now, you can likely predict the outcomes of further multiplications:

142857 * 4 = 571428 (starts with 5)

142857 * 5 = 714285 (starts with 7)

142857 * 6 = 857142 (starts with 8)

In essence, 142857 is a number that produces cyclic permutations of its digits when multiplied by integers 1 through 6. But what occurs when we multiply it by 7? Let’s explore that.

The Significance of 7

142857 * 7 = 999999

This is intriguing. Instead of cyclic repetition, we receive a sequence of six 9s. This result relates closely to the cyclic nature of 142857, which we will discuss shortly. But first, let’s examine a few more interesting attributes tied to this outcome.

If we split 142857 into two three-digit numbers and add them:

142 + 857 = 999

Fascinating, isn’t it? When we break the six-digit number into two three-digit segments, their sum yields three 9s.

What if we divide 142857 into three two-digit numbers and sum them?

14 + 28 + 57 = 99

This results in two 9s.

Now, suppose we cyclically divide 142857 into three four-digit numbers and sum them:

1428 + 5714 + 2857 = 9999

This gives us four 9s. The reason we added the four-digit numbers three times was to ensure every digit was used consistently.

Having explored the peculiarities with 7, let’s see what occurs with numbers greater than 7.

The Fancy Merry-go-round Continues

Let’s multiply 142857 by 8:

142857 * 8 = 1142856

Initially, this might seem complex, but if we take the last six digits and add them to the leftover digit(s), we arrive at:

1 + 142856 = 142857 (the cyclic number starting with 1)

We could extend this approach further:

142857 * 9 = 1285713

? 1 + 285713 = 285714 (the cyclic number starting with 2)

142857 * 10 = 1428570

? 1 + 428570 = 428571 (the cyclic number starting with 3)

And so forth.

When we multiply 142857 by 14, however, something different occurs:

142857 * 14 = 1999998

? 1 + 999998 = 999999

What’s so special about 14? Indeed, it’s a multiple of 7!

Now that we’ve explored some intriguing properties of the cyclic number 142857, let’s examine why it is cyclic in the first place.

The Mathematical Origin

Thus far, we have confirmed the cyclic nature of 142857 when multiplied by 1 through 6, and the anomaly that occurs when multiplied by 7. This oddity is closely tied to the cyclic properties of this number, and now we will uncover the underlying reason.

Let’s compute 1 divided by 7:

1/7 = 0.142857142857… (the first smallest digit appears after the decimal)

Indeed, this fraction repeats the cyclic number endlessly after the decimal. Next, if we multiply this by 10:

10/7 = 1.42857142857…

We can express 10/7 as a mixed fraction:

10/7 = 1 + 3/7 = 1.42857142857…

Subtracting 1 from both sides yields:

3/7 = 0.42857142857… (the third smallest digit appears after the decimal)

That’s an intriguing result! Let’s continue by multiplying this result by 10:

30/7 = 4.2857142857…

We can represent 30/7 as:

30/7 = 4 + 2/7 = 4.2857142857…

Subtracting 4 from both sides gives us:

2/7 = 0.2857142857… (the second smallest digit appears after the decimal)

You could keep multiplying the previous result by 10, and so on.

Ultimately, the reason 142857 is a cyclic number stems from the nature of fractions with 7 as the denominator. Conversely, you can express it as:

1/142857 = 0.000007000007…

If you’re still questioning whether 7 is special, it is not.

The Generalised Condition for a Cyclic Number

142857 is simply the first cyclic number. The next one is:

0588235294117647

This number contains 16 digits and originates from the fraction 1/17 = 0.0588235294117647. Multiplying by 1 through 16 results in cyclic permutations, while multiplying by 17 yields ‘9’ repeated 16 times.

What do 7 and 17 share in common? They are both prime numbers.

Let P represent a prime number and X symbolize an arbitrary cyclic number with any number of digits. Cyclic numbers emerge from primes in the following form (see Equation 1):

As demonstrated, the cyclic number infinitely repeats after the decimal point. When we multiply both sides by 10^(P-1), we get:

By subtracting Equation 1 from Equation 2, we arrive at:

This is the formal condition for cyclic numbers. Note that multiplying the right-hand side by P results in 9 repeating (P-1) times, explaining the unique outcomes.

However, not all primes fulfill this condition! The applicable primes are known as Full Reptend Primes, and a comprehensive list can be found here. Currently, no algorithms exist to compute full reptend primes.

Similarly, a list of all cyclic numbers can be found here. However, remember that all numbers beyond 142857 must be considered with a leading zero. To understand why, let’s revisit Fabiano Fagundes’s original comment:

“Mine is 142857. No kidding. I love the only cyclic number in the decimal system.”

He indicates that 142857 is the ONLY cyclic number in the decimal system. The rationale is that if we exclude numbers with a leading zero, 142857 stands alone as the only cyclic number in the decimal system. This fact indeed makes the number 7 special after all.

With that, I conclude this essay, and I hope you found it enjoyable.

A MAP of almost ALL of my work till date. Enjoy!

References and credit: Dr. Tony Padilla/Numberphile, Eric W. Weisstein, and Santanu Bandyopadhyay.