# Finite Periodicity of Pi

1. Dec 7, 2006

If I take the first p digits of pi, is it possible that what I will have is the first p/2 digits repeated twice?

For example, suppose pi = 3.14153141529485729487... and I took the first 10 digits and got 3141531415 which is 31415 repeated twice. Can this happen?

2. Dec 7, 2006

### AlphaNumeric

It's conceivable that there exists irrational numbers where the first n digits in the decimal expansion repeat themselves and then after 2n digits, it goes non-repeating, but that's not true with pi, at least to a trillion digits.

/edit

Infact, it's not just conceivable, it's pretty obvious!

$$\sqrt{2} = 1.4 \ldots$$ Therefore $$0.123123 + \frac{\sqrt{2}}{10^{7}}$$ is irrational, but has the first three digits repeat.

Last edited: Dec 7, 2006
3. Dec 7, 2006

Yeah, I knew it wasn't true for arbitrary irrationals because the first example I thought of was (101010+pi) which is irrational and repeats 10 to start. I guess I was looking for existence of evidence that it happens in pi or if there's some way to prove that this never happens for some number.

4. Dec 11, 2006

### Gib Z

Pi can been calculated to i think its 14 trillion digits? No pattern found, but there are yet no proofs as to why, or even that each digit occurs infinity in the decimal expansion...

5. Dec 12, 2006

### HallsofIvy

It is quite possible that for some n, the second n digits in $\pi$ are just the first n repeated but, if so, it must be some exceptionally large n. No one knows for sure but I would consider it unlikely.

I notice you titled this "periodicity of $\pi$" which implies a block of digits repeating over and over again, forever. Since $\pi$ is NOT rational, that certainly cannot happen.

6. Dec 12, 2006

By "finite periodicity" I meant that the first n numbers are repeated a finite number of times, so it may appear periodic if you only look at the first k*n digits.

7. Dec 12, 2006

### HallsofIvy

Ah. In that case, as everyone else has told you- possible, unknown, very unlikely! (Unlikely that the first n numbers repeat k times for some n, k. Highly likely that some set of n numbers repeats k times in a row but still unknown.)

8. Jun 27, 2011

### asmani

With the assumption of normality of pi, can we calculate the probability that this statement is true for some p?
I think for a normal number the probability is less than 1/9.