An Upper Bound for Locating Strings with High Probability Within Consecutive Bits of Pi

Numerous studies on the number pi ($\pi$) explore its properties, including normality and applicability. This research, grounded in two hypotheses, proposes and proves a theorem that employs a Bernoulli experiment to demonstrate the high probability of encountering any finite bit string within a sequence of consecutive bits in the decimal part of $\pi$. This aligns with findings related to its normality. To support the hypotheses, we present experimental evidence about the equiprobable and independent properties of bits of $\pi$, analyzing their distribution, and measuring correlations between bit strings. Additionally, from a cryptographic perspective, we evaluate the chaotic properties of two images generated using bits of $\pi$. These properties are evaluated similarly to those of encrypted images, using measures of correlation and entropy, along with two hypothesis tests to confirm the uniform distribution of bits and the absence of periodic patterns. Unlike previous works that solely examine the presence of sequences, this study provides, as a corollary, a formula to calculate an upper bound N. This bound represents the length of the sequence from $\pi$ required to ensure the location of any n-bit string at least once, with an adjustable probability p that can be set arbitrarily close to one. To validate the formula, we identify sequences of up to $n=$ 40 consecutive zeros and ones within the first N bits of $\pi$. This work has potential applications in Cryptography that use the number $\pi$ for random sequence generation, offering insights into the number of bits of $\pi$ required to ensure good randomness properties.