## Mathematical Beauties of the Number 142587

There are some mysteries that the human mind will never penetrate. To convince ourselves we have only to cast a glance at tables of primes and we should perceive that there reigns neither order nor rule.
Leonhard Euler

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The prime factorisation of the number 142857 is 33 × 11 × 13 × 37, and the divisors of the number are 1, 3, 9, 11, 13, 27, 33, 37, 39, 99, 111, 117, 143, 297, 333, 351, 407, 429, 481, 999, 1221, 1287, 1443, 3663, 3861, 4329, 5291, 10989, 12987, 15873, 47619, 142857.

Numbers are always beautiful and in this blog post let’s play with the beautiful number 142587! One of the occurrences of this number can be found in oneseventh (1/7). Let’s start this post with Midy’s theorem and discuss other interesting properties of the number in succession.

Midy’s Theorem If the period of a repeating decimal for a/p, where p is prime and a/p is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. In the case of 1/7 (oneseventh), the period is 6. In fact, we have

In this case, we have 142 + 857 = 999. Again, if we split the number 142857 into a group of two, and the groups are added, the result is 99: 14 + 28 + 57 = 99. Even, the sum is 1428 + 5714 + 2857 = 9999!

A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. A cyclic number has an unusually interesting property. If we multiply a cyclic number, by 1 through n (where n is the number of digits of the cyclic number), these products contain the same n digits of the initial number in exactly the identical cyclic order. Two examples are 76923 and 588235294117647.

Of course, 142857 is a cyclic number. Digits of 142857 always appear in the same order, but cycle around, when multiplied by 1 through 6, as shown below. In fact, the first digit of every product goes to the end of the number to form the next product. Otherwise, the order of the digits is intact. Moreover, the sum of the digits in each of these product numbers is equal to 27.

The product of the number 142857 and 7 generates the following result:

142857 × 7    = 999999

What if we multiply by an integer greater than 7 (but not a multiple of 7)? There is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits to the remaining digits and repeating this process (if required) until only six digits are left, it will result in a cyclic permutation of 142857. For example,

If it is a multiple of 7, then it always leads to 99999! For example,

Consider the multiplication results of 142857 by the numbers 1 to 6. Divide the result into two parts – each consisting of three digits. If we square these parts and subtract the smaller one from the bigger one, we always get back a cyclic permutation of the number 142857.

The product of the number 142857 and multiples of 7 generates the following set of results:

A few noteworthy observations are as follows:

(a) With an increasing multiplier of 142857, the leftmost digits of the product will be in ascending order, such as 1, 2, 3, and so on.

(b) As the multiplier of 142857 increases, the rightmost digits of the product will be in decreasing order, such as 8, 7, 6, and so on.

(c) When the sum of the multipliers is held constant (equal to 77), the products are reverse of each other.

142857 × 14 = 1999998

142857 × 63 = 8999991

142857 × 21 = 2999997

142857 × 56 = 7999992

142857 × 28 = 3999996

142857 × 49 = 6999993

142857 × 35 = 4999995

142857 × 42 = 5999994

In base 10, 1428572 = 020408122449. If we split it into two 6-digit numbers, 020408 and 122449, and add them up, we get 142857 back:

020408 + 122449 = 142857.

Thus 142857 is a Kaprekar number.

Is there anything interesting about this division? We can show that, 6 is the smallest value of n such that 10n – 1 is divisible by 7. In that case, $\frac{10^6-1}{7}$ = 142857 and we get

Let us unveil another interesting property of 142857 by inserting a 9 in the middle of the number. In this way, we obtain 1429857 from 142857. When multiplied by any number from 1 through 6, the product retains the cyclic nature of the original cyclic number, always maintaining a 9 in the middle, as demonstrated below. This fascinating property adds a layer of mathematical elegance to the cyclic nature of 142857 and its derivatives.

An enneagram is, literally, a drawing with nine lines. It consists of a circle with nine equidistant points on the circumference. The points are connected by two figures: one connects the numbers 1 to 4 to 2 to 8 to 5 to 7 and back to 1; the other connects 3, 6, and 9. The 142857 sequence is based on the fact that dividing 7 into 1 yields an infinite repetition of the sequence 142857.

Try These

Q1. Show that if n is a positive integer such that $\frac{1}{n}$ is periodic with period n – 1, then n cannot contain any factor of 2 or 5.

Q2. Show that if n is a positive integer such that $\frac{1}{n}$ is periodic with period n – 1, then n has to be a prime number.

Q3. Find the next five cyclic numbers after 142857

Note. It is still an open problem whether there are infinitely many cyclic numbers.

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