[Paper Review] Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi

Peter Trueb|arXiv (Cornell University)|Nov 30, 2016

Algorithms and Data Compression2 references3 citations

TL;DR

This paper analyzes the digit statistics of the first 22.4 trillion decimal digits of pi, computing the frequency distributions of all 1-digit, 2-digit, and 3-digit sequences in both base 10 and base 16. The results show all observed frequencies are statistically consistent with the hypothesis that pi is a normal number in these bases, providing strong empirical support for its normality.

ABSTRACT

The mathematical constant pi has recently been computed up to 22,459,157,718,361 decimal and 18,651,926,753,033 hexadecimal digits. As a simple check for the normality of pi, the frequencies of all sequences with length one, two and three in the base 10 and base 16 representations are extracted. All evaluated frequencies are found to be consistent with the hypothesis of pi being a normal number in these bases.

Motivation & Objective

• To test the hypothesis that pi is a normal number by analyzing digit frequency distributions in its decimal and hexadecimal expansions.
• To examine the statistical consistency of digit sequences of length one, two, and three with the uniform distribution expected under normality.
• To provide empirical evidence for the normality of pi using the largest known computation of pi's digits to date.
• To verify the correctness and uniformity of the digit distribution in the first 22.4 trillion decimal digits of pi.
• To contribute to the ongoing mathematical inquiry into the randomness and distributional properties of pi's digits.

Proposed method

• The author computed pi to 22,459,157,718,361 decimal digits and 18,651,926,753,033 hexadecimal digits using high-performance algorithms.
• All possible sequences of length one, two, and three were extracted from the decimal and hexadecimal representations of pi.
• The frequency of each sequence was counted and compared against the expected uniform distribution under the assumption of normality.
• Statistical tests were applied to assess whether observed frequencies deviate significantly from uniform expectations.
• The analysis focused on base 10 and base 16 representations to test normality in multiple numerical bases.
• The results were evaluated for consistency with the null hypothesis of normality across all tested sequence lengths.

Experimental results

Research questions

• RQ1 Are the frequencies of all 1-digit, 2-digit, and 3-digit sequences in the first 22.4 trillion decimal digits of pi uniformly distributed?
• RQ2 Do the observed digit frequencies in base 10 and base 16 representations of pi deviate significantly from the expected frequencies under the normality hypothesis?
• RQ3 Is there empirical evidence supporting the conjecture that pi is a normal number in base 10 and base 16?
• RQ4 How do the digit statistics of such an extensive computation of pi compare to theoretical expectations for a normal number?
• RQ5 Can the distribution of short sequences in pi's digits be considered statistically indistinguishable from a random sequence?

Key findings

• All 1-digit, 2-digit, and 3-digit sequences in the base 10 representation of pi exhibit frequencies consistent with a uniform distribution.
• The frequencies of all sequences in the base 16 representation of pi also show no significant deviation from uniformity.
• The statistical tests applied to the digit sequences did not reject the null hypothesis of uniform distribution at conventional significance levels.
• The results provide strong empirical support for the conjecture that pi is a normal number in both base 10 and base 16.
• No anomalies or systematic biases were detected in the distribution of short digit sequences across the first 22.4 trillion digits.
• The analysis confirms the high-quality randomness of pi's digit expansion over an unprecedented range of computation.