Can the First p Digits of Pi Repeat Twice?

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In summary, it is possible for irrational numbers to have the first n digits repeated twice, but it is highly unlikely for this to happen with pi. There is no proof that this never happens, but it is believed to be very uncommon. Additionally, the concept of "finite periodicity" where a set of n digits repeats a finite number of times is also possible, but the probability of this occurring is low, especially for normal numbers such as pi.
  • #1
LeBrad
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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?
 
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  • #2
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!

[tex]\sqrt{2} = 1.4 \ldots[/tex] Therefore [tex]0.123123 + \frac{\sqrt{2}}{10^{7}}[/tex] is irrational, but has the first three digits repeat.
 
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  • #3
AlphaNumeric said:
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!

[tex]\sqrt{2} = 1.4 \ldots[/tex] Therefore [tex]0.123123 + \frac{\sqrt{2}}{10^{7}}[/tex] 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.
 
  • #4
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
It is quite possible that for some n, the second n digits in [itex]\pi[/itex] 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 [itex]\pi[/itex]" which implies a block of digits repeating over and over again, forever. Since [itex]\pi[/itex] is NOT rational, that certainly cannot happen.
 
  • #6
HallsofIvy said:
I notice you titled this "periodicity of [itex]\pi[/itex]" which implies a block of digits repeating over and over again, forever. Since [itex]\pi[/itex] 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.
 
  • #7
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
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.
 

Related to Can the First p Digits of Pi Repeat Twice?

1. What is finite periodicity of Pi?

The finite periodicity of Pi refers to the fact that the decimal representation of Pi, which is an irrational number, eventually repeats itself after a certain number of digits. This repetition is known as periodicity and is a characteristic of all rational numbers.

2. How is finite periodicity of Pi calculated?

The finite periodicity of Pi is calculated by dividing the circumference of a circle by its diameter. This value, known as Pi, is approximately 3.14. However, due to its irrational nature, Pi cannot be expressed as a finite decimal and will continue infinitely without repeating.

3. Why is finite periodicity of Pi important?

The finite periodicity of Pi is important in many branches of mathematics and physics. It is used in calculating the area and circumference of circles, as well as in various formulas and equations in geometry, trigonometry, and calculus. It also has applications in fields such as engineering, astronomy, and computer science.

4. Is there a pattern to the digits in the finite periodicity of Pi?

No, there is no discernible pattern to the digits in the finite periodicity of Pi. Despite numerous attempts by mathematicians, no repeating pattern has been found in the decimal representation of Pi. This is what makes it an irrational number.

5. Can the finite periodicity of Pi be calculated to an exact value?

No, it is impossible to calculate the finite periodicity of Pi to an exact value. This is because Pi is an irrational number, meaning it cannot be expressed as a precise fraction or decimal. Its decimal representation will continue infinitely without ever repeating or ending in a pattern.

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