Remember back in school, learning how to convert fractions to decimals and decimals to fractions?

They seemed so pesky: there were the easy ones like 1/2 or 1/4 or 1/5 which when long-divided would finish off really quickly to give the results 0.5, 0.25 or 0.2. There were the slightly-more-difficult ones that repeated, but it was possible to quickly figure out which numbers were repeating... and then there were fractions such as 1/7, which caused problems on both fronts: its value was hard to memorise and if long-divided, came up with a sequence of numbers that was long enough for you not to realise it was actually repeating.

**Composition of the Magical Number**

However, if we take a closer look at this magical number, something else can be found. The number 1/7 is equivalent to 0.1428571....142857... and so on. If this repeating number is to be looked at alone, many interesting properties can be found.

The very structure of the number is a bit odd-looking. If looked at closely, the following pattern seems to reveal irregularity:

The first two digits 14, is 7 x 2,

The second two digits 28, is 7 x 4

...yet the last set of digits 57 are only 1 greater than the next number in the sequence 7 x 8.

**Cyclical Property**

However the number 142857 itself has its own pattern. It is a cyclical number. To demonstrate this property, examine the multiplication table for this number:

142857 x 1 = 142857

142857 x 2 = 285714

142857 x 3 = 428571

142857 x 4 = 571428

142857 x 5 = 714285

142857 x 6 = 857142

The digits in the tables never change! All the answers in the first six entries of the multiplication table have the digits 1, 4, 2, 8, 5 and 7. The next number is also a bit of a shock (notice that this occurs on the *7th* entry in the table):

142857 x 7 = 999999

When the number is multiplied by 7 it comes out at exactly one off a million! The pattern mentioned before, however, continues:

142857 x 8 = 1142856

142857 x 9 = 1285713

142857 x 10 = 1428570

142857 x 11 = 1571427

Although it may seem that the pattern has now been broken, even with the extra digit the pattern continues. If the last digit is added with the first, then the remaining digit appears, ie:

1142856: Extra digits 1 and 6 added give 7, the missing digit

1285713: Extra digits 1 and 3 added give 4, the missing digit

and so on.

This pattern continues in the entire 142857 multiplication table save that every 7th number is a few digits short of a million. Exactly how many digits the answer would be short of a million depends on where the multiplying number lies in the multiplication table of 7. For example: 142857 x 35 = 4,999,995. Exactly 5 short of 5 million. This is because 35 is 5 times 7.

Even the prime factors of 142857 have some unusual properties as well. Not only that all of these numbers when inverted produce repeating decimals, but they have patterns of their own. The prime factors of 142857 are:

3 11 13 37

**Patterns in number 3**

The number 3 as it is well known, has the unusual property that if any multiple of 3 taken, its digits would add up to another multiple of 3. Some examples:

27: 2 + 7 = 9, which is 3 x 3

48: 4 + 8 = 12, which is 3 x 4

81: 8 + 1 = 9, which is 3 x 3

or even

142857: 1 + 4 + 2 + 8 + 5 + 7 = 27, which is 3 x 9.

**Patterns in Number 11**

The number 11 has the property that when multiplying the sum of the exterior digits (note the number of digits varies depending upon how many digits there are in the number, for example when multiplying with a two-digit number then one each from the exterior matter, whereas a three-digit number means that two digits on each side must be taken. The number of digits to take is always one less than the length of the multiplying numbers, see shown examples) becomes the interior of the product, with the result carried over. Some examples:

11 x 15: 1 + 5 = 6, so end result is 165

11 x 135: 13 + 35 = 48, so end result is 1485

11 x 1345: 134 + 345 = 479, so end result is 14795

11 x 28: 2 + 8 = 10, so end result is 308 (one carried over)

11 x 288: 28 + 88 = 116, so end result 3168 (one carried over)

11 x 2888: 288 + 888 = 1176, so end result is 31768

**Patterns in Number 13 - The other Cyclical Producer**

The number 13, along with having a good amount of history, has its own cyclical produce. 1/13 produces the repeating digits 0.076923...076923...and so on, when the number 76923 is taken with the leading zero indicator it also behaves like a cyclical number. However, it is not a cyclical number as it does not permutate in consecutive multiplications. Some examples:

076923 × 1 = 076923

076923 × 3 = 230769

076923 × 4 = 307692

076923 × 9 = 692307

....^{1}

**Patterns in Number 37**

Finally, the number 37. This number also exudes a pattern in its multiplication tables:

37 x 1 = 37

37 x 2 = 74

The second number in the sequence is one off the anagram of the last number in the sequence, ie:

3 7 = 37

X

7 3 + 1 = 74

This is followed by triple ones:

37 x 3 = 111

and then...

37 x 4 = 148

37 x 5 = 185

37 x 6 = 222

37 x 7 = 259

37 x 8 = 296

The pattern continues, the fifth number is again one off the anagram of the last number in the sequence, followed by triple twos (note the repeating digits occur on every 3rd sequence instance). This pattern continues on and on even past the point 999. However, shortly within the sequence, the numbers begin to split like the sequence for 142857 itself and a bit of work is needed to see the pattern:

37 x 9 = 333

37 x 10 = 370

37 x 11 = 407

Notice how the extra one that we were adding before has been added to the three instead; however, the composition of the result would remain the same be it that the order might change.

The exact usage of these patterns has yet to be determined. Who knows - it may be a reader of this Entry who takes it that step further?