An Upper Bound for Locating Strings with High Probability Within Consecutive Bits of Pi
Numerous studies on the number pi (<inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>) explore its properties, including normality and applicability. This research, grounded in two hypotheses, pro...
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2025-01-01
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| author | Víctor Manuel Silva-García Manuel Alejandro Cardona-López Rolando Flores-Carapia |
| author_facet | Víctor Manuel Silva-García Manuel Alejandro Cardona-López Rolando Flores-Carapia |
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| description | Numerous studies on the number pi (<inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>) 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 <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>, 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 <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>. 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 <i>N</i>. This bound represents the length of the sequence from <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> required to ensure the location of any <i>n</i>-bit string at least once, with an adjustable probability <i>p</i> that can be set arbitrarily close to one. To validate the formula, we identify sequences of up to <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>=</mo></mrow></semantics></math></inline-formula> 40 consecutive zeros and ones within the first <i>N</i> bits of <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>. This work has potential applications in Cryptography that use the number <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> for random sequence generation, offering insights into the number of bits of <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> required to ensure good randomness properties. |
| format | Article |
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| language | English |
| publishDate | 2025-01-01 |
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| spelling | doaj-art-7103ee1cd4814e22b46000c192f6d0482025-01-24T13:40:09ZengMDPI AGMathematics2227-73902025-01-0113231310.3390/math13020313An Upper Bound for Locating Strings with High Probability Within Consecutive Bits of PiVíctor Manuel Silva-García0Manuel Alejandro Cardona-López1Rolando Flores-Carapia2Centro de Innovación y Desarrollo Tecnológico en Cómputo, Instituto Politécnico Nacional, Mexico City 07738, MexicoCentro de Investigación en Computación, Instituto Politécnico Nacional, Mexico City 07738, MexicoCentro de Innovación y Desarrollo Tecnológico en Cómputo, Instituto Politécnico Nacional, Mexico City 07738, MexicoNumerous studies on the number pi (<inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>) 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 <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>. 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 <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>, 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 <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>. 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 <i>N</i>. This bound represents the length of the sequence from <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> required to ensure the location of any <i>n</i>-bit string at least once, with an adjustable probability <i>p</i> that can be set arbitrarily close to one. To validate the formula, we identify sequences of up to <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>=</mo></mrow></semantics></math></inline-formula> 40 consecutive zeros and ones within the first <i>N</i> bits of <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula>. This work has potential applications in Cryptography that use the number <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> for random sequence generation, offering insights into the number of bits of <inline-formula><math display="inline"><semantics><mi>π</mi></semantics></math></inline-formula> required to ensure good randomness properties.https://www.mdpi.com/2227-7390/13/2/313Bernoulli experimentchaosentropynormal numbersPi numberupper bound |
| spellingShingle | Víctor Manuel Silva-García Manuel Alejandro Cardona-López Rolando Flores-Carapia An Upper Bound for Locating Strings with High Probability Within Consecutive Bits of Pi Mathematics Bernoulli experiment chaos entropy normal numbers Pi number upper bound |
| title | An Upper Bound for Locating Strings with High Probability Within Consecutive Bits of Pi |
| title_full | An Upper Bound for Locating Strings with High Probability Within Consecutive Bits of Pi |
| title_fullStr | An Upper Bound for Locating Strings with High Probability Within Consecutive Bits of Pi |
| title_full_unstemmed | An Upper Bound for Locating Strings with High Probability Within Consecutive Bits of Pi |
| title_short | An Upper Bound for Locating Strings with High Probability Within Consecutive Bits of Pi |
| title_sort | upper bound for locating strings with high probability within consecutive bits of pi |
| topic | Bernoulli experiment chaos entropy normal numbers Pi number upper bound |
| url | https://www.mdpi.com/2227-7390/13/2/313 |
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