I discovered that the decimal representation of 15/17 consists of an endlessly repeating sequence of 16 digits. I wonder what rational number has the longest sequence of repeating digits?

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- Jul 22nd 2010, 02:30 PMStevenBrownRepeating decimals
I discovered that the decimal representation of 15/17 consists of an endlessly repeating sequence of 16 digits. I wonder what rational number has the longest sequence of repeating digits?

- Jul 22nd 2010, 02:36 PMStevenBrown
Thinking more about this, it occurs to me that a rational number expressed as a fraction having arbitrarily long numerator and denominator that differ by 1 can have an astronomically long sequence of repeating digits in the decimal representation.

- Jul 22nd 2010, 04:04 PMwonderboy1953
- Jul 22nd 2010, 04:14 PMundefined
You might find useful this reference (also the other subheadings)

Wikipedia - Repeating decimal - subheading Other properties of repetend lengths

You might also like to think backwards; given any positive decimal expansion with a repetend, we can find unique positive integers p,q such that gcd(p,q)=1 and p/q has the given value. - Jul 22nd 2010, 06:03 PMStevenBrown
It occurred to me that a repeating decimal such as 15/17 =

0.8823529411764705 8823529411764705 8823529411764705 ...

is "going fractal," in the sense that the same pattern repeats endlessly on smaller and smaller scales. - Jul 22nd 2010, 09:21 PMSoroban
Hello, StevenBrown!

You discovered this on your own? . . . Good for you!

Keep exploring . . . there's some fascinating stuff out there!

Quote:

I discovered that the decimal representation of

consists of an endlessly repeating sequence of 16 digits.

I wonder what rational number has the longest sequence of repeating digits?

Considerable exploration has already been done on this subject.

Here are some basics that I remember . . .

If is a prime, the decimal for has a digit repeating cycle.

However, some primes can have even shorter cycles.

For example: .

What determines the length of the cycle for prime reciprocals?

It is the least for which is divisible by

In baby-talk, consider a string of 9's.

Begin dividing by a prime

When it "comes out even", stop.

The number of 9's is the length of the cycle.

Example:

The first time the division stops is: .

. . Therefore, has a 6-digit cycle.

Example:

The first time the division stops is: .

. . Therefore, has a 3-digit cycle.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Here's another interesting feature . . .

. .

The last five decimals are cyclic arrangements of the first one.