Can the First p Digits of Pi Repeat Twice?

[LeBrad]

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?

 

[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.

 

[LeBrad]

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.

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.

 

[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...

 

[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.

 

[LeBrad]

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.

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.

 

[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.)

 

[asmani]

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?

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.